OK, so first a clarification of the contexts:
The quoted materials are mostly concerned with Western European music. The music of the Middle Ages almost certainly refers to the liturgical music of that time and place, so we're mostly talking about Gregorian Chant, organum, motets, etc. When talking about changes in the later eras, I suspect they're mostly talking about changes that had entirely solidified by the time of Classical composers like Mozart and Beethoven. If we push further into the twentieth and twenty-first century music of that lineage we see further changes in interval usage, as we would if we push into other genres such as jazz, rock, pop and hip-hop.
Next, an executive summary:
I suspect that the first book is mostly referring to a shift from the perfect fourth being mostly treated as a consonance in the Middle Ages to being treated as a dissonance in some contexts in the Classical era. Concomitant with this shift is a shift from a tendency to treat what we now call imperfect consonances (major and minor thirds and sixths) as dissonances in the Middle Ages to the consonances they generally are in the Classical era. We could also speak of a much later shift from Classical to Jazz: in the former the seventh is virtually always a dissonance while it's often a consonance in the latter.
To go into more detail, I think it's extremely useful to shore up the sometimes loose definitions of "consonant" and "dissonant." In particular, I think it would be helpful to use the terms "concordant" and "discordant" to refer to objective measures of how simply the frequencies of two notes relate to each other. This is shown somewhat in the picture you posted that includes an illustration of the highly concordant interval of the octave—wherein the periods of the two waveforms are highly correlated—and the fairly discordant interval of a second. The waveforms of the second take far longer before they get back to their starting relationship (not shown in the image, since the image would have to be too long for the page width) than the octave, which shows the waveforms getting back to their starting relationship twice in just a brief span. Although there might be some ambiguities and debate about a highly-specific ranking of every possible interval, it's fair to say generally that perfect unisons and octaves are extremely concordant; perfect fourths and fifths and quite concordant; major and minor thirds and sixths are moderately concordant; minor sevenths and major seconds are fairly discordant; major sevenths and minor seconds are highly discordant. This is purely based on how the pitches interact with each other in terms of their periodic waveforms.
If we let those terms stand for a more "scientific" understanding of intervallic relationships, we can then use "consonant" and "dissonant" to refer to the way that composers (in the broadest sense of that term) actually use the intervals within the context of their piece and genre. A dissonant interval is one that tends to be treated as unstable: an interval that needs to resolve to a more consonant interval. A consonant interval is one that is treated as stable; it's an interval that can be resolved to. Because this is entirely about style and context, this is the aspect of interval definitions that has shifted the most over the years. Unfortunately, there's one more complication that has to be discussed before talking about that.
For the most part, introductory texts treat pitch and tuning as far clearer than they actually are. They tend to assume equal temperament, which is fine for beginners, but the question of historical shifts in interval usage cannot be separated from the issue of temperament. There are lots of questions on this site that will give you the details, but suffice to say that the earliest tunings used during the Middle Ages for fixed-pitch instruments (instruments like fretted strings or keyboards where you have to set the exact tuning of the pitches in advance and can't easily modify the tuning during performance) was Pythagorean. This is a system where all the notes are tuned according to perfect fifths. If you do this—essentially tune your piano by going around the circle of fifths and getting every fifth truly perfect—then your major thirds will be quite discordant compared to our modern equally-tempered major thirds and will be very discordant compared to a pure major third. Again, the specifics of these differences are complicated, but you can find out more by looking up "just intonation", "temperament" and "equal temperament" among other topics. The upshot as far as this question is concerned is that it isn't surprising that composers of liturgical music in the Middle Ages treated the major third as a dissonance that needed to resolve to purer intervals such as the fourth and fifth. In part this is because it is less common to treat an interval as discordant as the Pythagorean major third as a point of resolution.
It's vastly oversimplified, but very generally speaking, different attitudes start to emerge (over the course of centuries) during the Renaissance. The sweetness of pure thirds (which were easy to explore in a capella singing for example) became more popular, and people started tuning fixed-pitch instruments so that they had pure (or purer) thirds as well. Composers started using thirds (and sixths) as primary consonances and so needed to adjust the tuning of fixed-pitch instruments to make those thirds more concordant as well. As the third became a primary consonance, an ambiguous status for the perfect fourth emerges. Although the perfect fourth was just as concordant as it was before (more concordant than a major third!) it was only a half step away from being a third. This led to a natural "gravity" as—in the context of the music that was being written—the top note of those fourths tended to "want" to resolve down a half step and become a consonant third. The objective concordance of the perfect fourth didn't change, but its contextual meaning did. When the bottom note of the fourth is the lowest note of an entire harmony, the fourth became a clear and strong dissonance.
The fourth didn't become more discordant, it just became more dissonant.
And, just to reiterate, this is only in certain contexts even within music of the Classical era. When the fourth is between two upper notes within a larger harmony, it's perfectly consonant. The fourth is only treated like a dissonance when the bottom note of the fourth is also the lowest note of the entire harmony. (Look up the cadential 6/4 chord for the quintessential example of this, although also see the conversation in the comments to this answer between me and @leftaroundabout, which adds some interesting subtleties to this issue). The distinction between concordance/discordance on the one hand and consonance/dissonance on the other also helps to clarify your question about the augmented second versus the minor third. First off, in many non-equal tempered scenarios, those two intervals are not the same. The augmented second will be wider and far more discordant than the pure minor third. The books you're quoting, however, are talking about equally-tempered instruments like the modern piano, and on those it is true that the augmented second and minor third interval are the same objective relationship. As a result, it would be entirely correct to say that they are equally concordant (or discordant), but because "consonance" and "dissonance" are contextually defined, it isn't ridiculous to say that an augmented second is dissonant while a minor third is consonant. That tends to be their usage in music of the Classical era: a composer would almost never spell out the interval as an augmented second unless they intended it to be performed as an instability that needs to resolve. The spelling of notes in the score is not simply an instruction about what buttons to push, it can also indicate the rhetorical intent of the composer.
So, to sum up, the fourth was initially treated as a consonance equal to the perfect fifth in the Middle Ages of Europe and thirds tended to be dissonances that resolve to them. As the status of thirds began to change, the fourth became ambiguous and is often a dissonance that needs to resolve to a third. Historically, there was some shift in the concordance of thirds due to changing standards of tuning, but the primary shift is a purely contextual one toward consonance. More recently, we hear something similar happen to the treatment of the seventh in jazz music, where it's often a perfectly consonant interval despite its relative discordance.
EDIT TO ADD:
I just saw where you asked for sources. One problem is that this kind of question falls into an unfortunate hole in the literature. On the one hand, it's too basic for discussion in academic journals; on the other hand, the historical evolution of something so specific is seen as too tangential for an introductory text. I think most textbooks of both music theory and music history at least mention it, sort of like the book that prompted your question. I know that both the Clendinning/Marvin and Kostka/Payne music theory texts at least devote a paragraph to the topic. I don't have my copy on hand, but I think there are a few paragraphs in the Burkholder/Grout/Palisca textbook on this as well—however I recall that is just discussed across the course of the text as it moves from early music into the Baroque, not in one clear place.
Perhaps the best place to look for authoritative sources is in Grove Music (now called Oxford Music). The article on the Fourth in there is pretty brief, but includes this paragraph:
The 4th has a unique position in Western music because it has been regarded as a Perfect interval (like the unison, 5th and octave) and a dissonance at the same time. In ancient Greek music the basis of melody was the Tetrachord, a set of four pitches encompassed by a 4th. The earliest forms of medieval parallel Organum favoured it as the interval between the vox organalis and vox principalis. With the further development of polyphonic music in the 12th and 13th centuries, the 5th replaced the 4th as the most important Consonance after the octave and the unison. By the 15th century the 4th appeared as a consonance only between the upper parts of a vertical sonority, for example in 6-3 chords of the fauxbourdon style and at 8-5-1 cadences (e.g. D–A–D); composers of the later 15th century, including Du Fay, sometimes deliberately avoided the 4th in three-part writing (see Non-quartal harmony), and Tinctoris deemed it a dissonance in his Terminorum musicae diffinitorium (c1473).
You'll notice that the concord-discord/consonance-dissonance distinction I talk about is far from universal, and the terms are used differently in most texts on the topic. Ultimately, this is the kind of information I learned during a semester-long graduate course on The History of Theory. As is common in such courses, there was no textbook, we engaged with primary historical sources. We began with ancient theorists like Aristoxenus and, later, Gaudentius, then moved to later writers like Boethius et al. For the most part, these writers don't talk much about the shift in thinking, instead, one has to intuit and observe the shift between them. Anyway, I hope this at least helps situate your understanding a bit.
