How are things like ている/でいる split in terms of pronunciation? Do people internalize/think of it as te-iru or tei-ru when speaking? (there is a small difference in what it would sound like between those two) Or would it depend on the meaning? (Like 食べている vs 帰っている?)
Grammatically, that いる is a subsidiary verb that follows the te-form of a verb.
So it's 食べて-いる, not 食べてい-る. If you are a beginner you can have a small pause between て and いる when reading.
Native speakers say this almost like a single word. Regarding how native speakers split sentences when reading very slowly, see the following questions (but please don't go overthink it).
Basically, "ている" and "でいる" are always pronounced as te-iru and de-iru respectively. However you can hear "食べている" as tabeteiru and "帰っている？" as kaetteiru? without any split.
Actually we often say not "帰っている？" but "帰ってる？kaetteru?"
These variations of pronunciation don't make difference in their meaning.