TL;DR depends on the song. Sometimes one is obviously better than the other, but most of the time it's somewhere in the middle and both interpretations are valid.
This is often called a "line cliché", and depending on how it's executed it can sound either like harmonic movement or just like a moving baseline with "pedal notes" in the top. It can be completely chromatic, or it can include whole steps too. It occurs in major and minor keys (C, CΔ7/B, C7/B♭, F/A, Fm(6)/A♭ is pretty common in wishy washy love ballads too haha)
Often how to notate it (or think of it) isn't really a case of "this or that" but of perspective, both things are true, but different songs lean in different directions.
Are you trying to create an effect of constant harmony with a shifting bass, or are you trying to give a sense of harmonic movement. Or from a listener's perspective, which one does it sound more like? Generally, if it sounds like everything in the song is harmonically static, with just a wandering bass, then that's probably the best way to transcribe it. If it sounds more like a "chord progression", then it's best to write it like that.
A lot of it comes down to voicing, often in more "harmonic movement" sounding implementations of the line cliché, the voicings of the rest of the band change to accomodate the harmonic movement. (so in a band, the guitar for example won't just keep playing the same chord, but play different chords in addition to the bass playing different notes. Or on a piano, both the left and the right hand will change, not just the left hand.)
So let's say we have a common minor line cliché (just one example, there are many more)
1 Am, Am/G, Am/F# Am/F
2 Am, Am7/G, Am6/F♯ Am/F
3 Am, Am7/G, F♯ø, FΔ7
These are all ways of writing the essentially same chords. Generally a guide for me is whether it's literally only the bass that's changing, or whether there are harmonic changes throughout the range (in the "left hand" and "right" hand in piano terms, but in a whole arrangement, what are all the other instruments doing?")
Generally for me "1" implies a very rigid arrangement with only a moving bassline, whereas the 3rd one implies proper solid harmonic movement (that may include other extensions and more interesting voicings). If something is in the a "there's a bit of harmonic movement but it's only really the bass" then I'd lean towards the 2nd option; hedging my bets.
The other thing is how much does the arrangement tolerate the clashes caused by a line cliché. In the example I gave above, there were none; but when it's a pure chromatic line cliché, there's often a clash caused by that second chord. So Am, Am/G♯... for example. In fast more animated pop music (and often in pop music), you often just basically ignore it, keep playing A minor in the top and treat the ensuing weird chord as just a passing chord to the next note. On the other hand, in slower music (or in musical styles or bands with more of a focus on harmony), you'll find there will often be changes in the harmony to accompany that major 7th. E7/G♯, G♯dim, G♯+. The first two of these are very different from Am/G#, and so really need to be written.
But even in this case, there's still a middle ground there too, sometimes you don't really change the harmony (like with a G♯dim chord), you just try and avoid playing any As so there's no clash, and so you get an aug chord almost by accident. (i.e. ACE goes to G#CE). In that case I'd write it as G♯+ or G♯Aug, but honestly maybe in a pop context I'd instead write "Am/G♯"(no root) just to not scare people who don't really feel comfortable with Aug chords.
This answer could do with some refining, I'll pop back and clean it up later and maybe add some examples. Hopefully I've basically answered your question though.