apud means "near X", "at X", "next to X"
ĉe means "at X's place", "near but not next to X", "while encountering X", "at X but not physically"
Here are examples for ĉe from Wiktionary:
Ŝi sidas ĉe la fenestro. – She is sitting near the window.
ĉe malbona vetero – during bad weather
labori ĉe lumo de lampo – to work in light of a lamp
Li sidas ĉe la tablo. – He is sitting at the table. (this one is given using apud in this course, so there's some overlap in the meanings)
ĉe la marbordo – at the sea coast
Li parolis ĉe la kongreso. – He spoke at the congress.
Li estas profesoro ĉe universitato. – He is a professor at a university.
Ni renkontiĝu ĉe vi. – Let's meet at your place.
Li estas tre ekkolerema kaj ekscitiĝas ofte ĉe la plej malgranda bagatelo. – He is very impulsive and he often gets excited at the tiniest trifle.
Kiam li estis ĉe mi, li staris tutan horon apud la fenestro. – When he was at my place, he was standing at the window for a whole hour.
I agree. that “ĉe” is not specific enough for it to have to mean "home" rather than "house". After all, the given correct answer was "At your place I am doing well".. However, it occurs to me that a person can own houses that he doesn't live in, and that is not the meaning here, so perhaps "home" is better.
I don't know you, so is that from your experience as a long-time esperantist or based on your experiences speaking English? Because I don't know Esperanto well enough to comment on this usage of "ĉe," but other comments here demonstrate similar constructions in a few European languages, so it doesn't seem like a stretch to see that construction in Eo as well, even though we don't have it in English