Related: Any way to send a personal message to another user?
There isn't any way to do this other than by either commenting or posting in Chat. While participating in Chat requires 20 rep (so you can do that), and commenting anywhere requires 50, anyone can comment on their own posts.
A "Thanks for the edit!" comment could be flagged as too chatty, and you're right that it's not really worthwhile even as a thumbs-up. Because of that, a single post in Chat is the SE-friendly way of doing this. That's basically what Chat was provided for, as the linked MSE post explains.
Comments do have a feature which might be useful here, though.
The first few words of a comment appear in the recipient's inbox, as a sort of "teaser" with a link to the comment. This inbox message persists even after the comment is deleted. So it would be possible to write a comment, wait five or ten minutes for the system to pick it up and put a message in the inbox, and then delete the comment. If the comment consisted only of "Thanks for the edit. Really useful" then all of that should be visible in the inbox, even though it's no longer visible on the post itself.
If the comment is flagged as "too chatty" in the short time it's visible, then you deleting it will clear the flag. And if you forget to delete it yourself, a moderator will do so when the flag is dealt with. [So please don't forget!]
All that said, it's perfectly acceptable not to comment at all. It is nice to get an author saying "Thank you," but because there isn't really a convenient way of doing that, no-one expects to get thanks when they edit a post.