First of all: variation is part of our nature. Not only we're not able to do the exact same thing twice, but we often don't really like it.
And what you hear is not "off-beat".
There are two important aspects to consider:
- Before mechanical recording and reproduction of music were invented, there was no such thing as "the original version". Every rendition of every music piece was different, sometimes dramatically.
- Fixed tempo during the whole duration of the piece and specific metronome markings ("BPM") are relatively new in music history, especially for popular music that is commonly listened to by most people.
Before notation was standardized, music was only shared by ways of reproducing it by ear. This obviously meant that there were some "common" pieces that were known by many people, but each rendition was different due the transformation processes caused by every musician playing it in a slightly different (and personalized) way and the others trying to reproduce it, possibly with different instruments.
Even after notation became more common, a lot of music included parts that explicitly left some amount of freedom to the performer (cadenzas, ornaments), while others completely depends on the feeling of the performers: especially tempo, dynamics, duration in dynamic variations. Other aspects that might impact on those variations are technical: depending on the instrument, the acoustics of the place in which it's played.
If you take a single classical music piece, you won't be able to find two exactly identical renditions, from different musicians, and even from the same one.
Then, fixed-tempo-music is relatively new: while it was already common even in ancient "popular" (folk) music, that was mostly due to the fact that they were based on simple forms and/or intended for dancing (which clearly requires a steady tempo). Most classical music has time variations throughout the whole duration, and those variations are not always the same, depending on... interpretation.
Finally, metronome markings are relatively new: the metronome was invented at the beginning of the 19th century, before that there were only text indications (mostly in Italian: "Allegro" for "pretty fast", "Adagio" as "slowly", etc.) that were very approximate. Even after the metronome, most music didn't indicate any precise tempo marking, in order to leave some level of freedom to the... again, interpretation.
When soloists come in place (temporarily or for the whole piece) have even more freedom in these aspects. Even if the accompaniment is pretty "stable", their tempo is not always on a "fixed grid". That's because interpretation means allowing the performer to emphasize in different ways what they're playing/singing: sometimes it's virtuosity (imagine a singer that can keep a high note for a very long time), or because they want the audience to pay more attention to some aspects of a song/piece/word.
And, yes, sometimes it's for boredom, but in reality it's to avoid it, as playing the same thing in the same way for hundreds or thousands of times is really boring, and it's something that can be often perceived by the audience: if I'm bored, the audience can become bored too.
Singing is probably the practice that mostly takes advantage of the freedom of interpretation. Since we're very accustomed to human voice, as listeners we can perceive a lot of (different) things depending on how something is spoken (sung), and a good performer is able to play with those infinitesimal aspects, not unlike an actor can say the same line in dozens of different ways, each one of them passing different feelings to the audience.
In the beginning of this post I wrote that we, as humans, often don't like repetition.
It might seem counterintuitive: most people, nowadays, prefers to listen to the same piece again and again, and when they listen to it somewhere else (like, in live concerts), they are more likely to enjoy the piece if it's "the same" as they were used to. This is because our brain "enjoys" recognizing schemes we already know.
We are now accustomed to the fact that we can listen to the exact same piece again and again, many people enjoy this, you specifically express that:
In general I think the result is awkward and spoils my fun listening.
In fact, many people are actually bored to listen to the same exact version of the same song again and again, and really enjoy to listen to different renditions even from the same performer, and even if we really love "the original". That's also a reason for which many people prefer going to concerts: it's often an occasion to listen to something they like, but with a different "touch", and sometimes even get surprised. It's a way to find and realize new things, consider songs from new and different aspects, know the performers better, their feelings, their abilities.
That is strange, because usually the original was very successful for a reason.
There are a lot of reasons for which a piece is successful (including the fact that it's very popular and people listen to it very often in tv/radio/commercials/etc, and sometimes this results in a song that can become enjoyable even by those that originally didn't like it).
But, generally, it's not because the song was sung exactly in the way it was performed in the record as opposed to different ways. It just became successful (possibly also because it was sung in that way), but singing/playing it with a different interpretation doesn't mean that it's less "powerful" or enjoyable, or that they don't consider the aspects that made that song successsful.
On the contrary, there's music for which the live version is sometimes more popular than the original studio recording.
I was going to write this in comments, but I realized that the reply was going to be too extensive.
There are two main meanings for "off-beat": the common meaning, usually applied to syncopation, is when notes (or accents) are not following the strong accents of the bar (playing 2 and 4 in 4/4). Then there's what happens if the rhythm is completely off (incoherent) with the standard mathematical subdivisions, and it's generally the case of unexperienced (or just bad) musicians that are not able to follow tempo.
What you're referring to is not strictly off-beat, but one of the following: rithmic (and melodic) variations on the "original" you're used to, or phrasing, which is something that is more "slightly outside the time grid" than actual "off-beat" in musical terms.
The first is just the liberty of the performer, who can decide to slightly change something to their own taste (as said, we don't like to do the exactly same thing every single time) without completely change the core aspects of the piece. You may not like it, but it's part of the very nature of music, which is not a fixed sum of dots and lines on paper, nor its mechanical reproduction.
Phrasing is even more fascinating, and is a very broad aspect that relies on interpretation and perception of time, and is closely linked to "humanization".
While we're now accustomed to the concept of fixed, stable, precise tempo, human expression actually follows "waves" of tempo variations, even when playing together; that's why programmers of music software and electronic instruments spend a lot of time on "humanizing" algorithms: they make "mathematically precise" music sound more natural. In fact, even the studio versions you shared have some of these "off-beat" cases (like Diamond's at ~1.25): these are "windows" (sometimes simple notes, others entire phrases) in which the soloists have some level of freedom to stretch or expand tempo.
But don't consider this as something that is "not in tempo". Obviously there are many occurrences in which being "more in sync" is required, depending on various aspects, but music (luckily) is not always that strict. All of this is not "off-beat", it's just what music is and has always been.
If I may, I suggest you to try to overcome the feeling you have for these "different" versions: don't be "distracted" in the wrong way, as these distractions are often hints to more interesting things. There is a whole new world in every different performance, and it's usually more interesting than always listening to the same one. You already know that very well, don't you like to try something different occasionally? ;-)