A standard way to count rhythm across bars is to pick a count for the beats in a bar and then use the first beat as the bar count. So in 4/4, counting quarter notes, one counts:
One Two Three Four | Two Two Three Four | Three Two Three Four | ...
As subdivisions are added, typically syllables are used instead of numbers, such as "and" for eighth notes. So:
One and Two and Three and Four | Two and Two and Three and Four | ...
For triplet meter, one counts "One and a Two and a" etc. And for 16ths a typical pattern is "One e and a Two e and a" etc.
One can also imagine adding some indication of where one is in the form to the count, etc. For songs with simpler form or with clear lyrics it isn't too bad, but I'd love to have a way to keep a mental count that makes it difficult to screw up where I am in the form.
This method reuses both the number and certain syllables for more than one purpose. E.g. "Two" can mean both the second bar and the second beat while "and" can be either the off beat eighth or the first part of a triplet. Usually this isn't a problem, but I'm finding it a bit taxing as things get more complicated. E.g. at fast tempos I'll count "Three" and then sort of think "was that the third bar 'Three' or the third beat 'Three'?" I've mostly gotten to the point where that doesn't throw me off, but it seems like it is taking effort. Counting a triplet rhythm where there's also something exactly in the middle is quite vexing.
I've talked to musicians who use a pattern of sounds, e.g. "dig ga da" for triplets, and then either just know where they are in the song or use the chord progression and lyrics to keep track. For specific patterns there are mnemonic devices of using words or phrases to get the feel of the rhythm, e.g. the teach rhythm with food approach. Neither of these seems like a really robust tool.
Now I'm pretty sure someone somewhere has put thought into this and designed a system that tries to solve these issues. I don't really want to overthink this and I'm well aware a lot of it is just practice, but I'd love to hear how others approach trickier counting or, ideally, read a writeup of a really well thought out system for counting complex patterns.
Searches here and on Google have turned up relatively little, but perhaps I'm just using the wrong words. Anyone have thoughts or pointers?
Edit: per xerotolerant's answer, the search term I was looking for is "rhythmic solfège." There's actually a pretty good Wikipedia page, which I'm surprised I didn't find originally.