Kelso, Mac described very well how it works in comping. As far as your question about what the composer had in mind, I can only answer for myself:

When I compose, I generally "over-write" at first, specifying every nuance of a chord so it will be precisely the way I want it. Then I step back to see if I could instead use alt or something simpler. Usually I can, unless there is a particular melody note that requires a specific voicing without clashing, as Mac mentioned.

Otherwise, yes, it's up to you.

I don't play piano or guitar, so I try to adapt to the preferences of whomever I'm writing for. Some prefer I write the specific chord voicing, but most do not. They would rather see just a seventh chord, and they will decide whether to add the ninth, thirteenth, sharp eleven, vary the ninth or fifth, etc. I just trust to their skill and experience and get pleasantly surprised.

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