I'm also interested in difference. I'll ask my Ukrainian friend tomorrow. For my intuition, I'm Polish, хата sounds like Polish chata that means old cottage house. Also used in some regions as synonym to dom (дім) or by some young people as some kind of slang. Maybe that's some clue
It is a bit close to the German "e" (you mean as in "Leben" or "gehen", for example?) than to "i". It is something in between (sort of).
If you really just say that German "e" for "и" that is actually OK! It will sound like spoken Ukrainian, I think this kind of effect is especially pronounced in some western Ukrainian accents. We even have phonetic symbols for these sounds: ие and еи
So you could transcribe будинок as: