IMO, the reason why people have big difficulties understanding music in minor keys comes from two mistakes:
- MISTAKE #1: Instead of playing (repeating) lots of example music they want to know theory first, and
- MISTAKE #2: Having got ideas from theory (caused by mistake #1), they assume that they can select one single scale in which they can operate "diatonically" i.e. without any alterations to the scale needed, for example the Harmonic Minor scale. Just like they were able to do for major tonality.
First of all, I'd suggest first playing lots of songs in minor to learn concrete examples. Spend at least a few weeks doing that. And only after you have some hands-on experience, you try to look at how the examples you know could be described in theoretical terms.
And about the "staying in one scale", it doesn't work like that. It's better to consider for example the dominant chord introducing a chromatic alteration, as you say. When you're at the tonic chord, let's say, Am, the assumed scale is A natural minor. No G# there. But then at some point comes an E major chord, and then the G note is sharpened. Songs are not "in harmonic minor" all the time.
Trying to avoid chromatic alterations is a Bad Idea and goes against the way music works, and it imposes an unnecessary obstacle to understanding even simple songs in minor keys (i.e. minor tonics). IMO, accidentals and temporary alterations should be introduced from day 1, and not try to think that music is inside a scale. Scales are there only as helper grids, so you can reason about where the sounding notes are relative to basic default positions. Songs are not "in a scale", except for some very specific modal music where chromatically altering the scale would break the modal feeling. But minor tonality is not a mode.
Harmonic and melodic minor scales can be used modally, but that's more of a modal jazz thing.
What comes to the picture, I don't know who made it or how it was presented where it was used. But to tweak the picture so that it makes some sense, I added some annotations:
During the dominant chord, the 7th scale degree has to be raised or otherwise it's not a proper dominant chord. It's a "chromatic alteration" or something. This is the basic thing to do in a minor key. Trying to use the harmonic or melodic minor scales modally is not a basic thing.
Side-note. What causes the theory-first mentality? Is it social pressure and demand of legitimacy? Seemingly scientific theoretical and informed logical thinking is highly valued. Is it perhaps a desire to retain a feeling of "being in control", and to avoid childish - and therefore socially unacceptable - babbling and tumbling where you don't know what you're doing? An opposite but related phenomenon is wanting to learn to play pieces exactly note-for-note "correctly" and precisely, for example with these Synthesia videos with note blocks falling on keyboard keys. What's wrong with repeating, babbling, toying, playing, interacting with others? That's how you learn languages, and music is a language. In my opinion, a better formula for learning music would be something like this:
- 45% repeating examples
- 45% messing around with the examples, making changes and trying variations, making discoveries
- 10% theory, to identify patterns and to get names for the discovered patterns, forming concepts to assist communication and reasoning.