The options should probably include "chili" and "chili pepper" in addition to "hot chili." This isn't consistent between the answers, but all are correct in the vernacular.
Azeeʼdíchʼííʼ simply means chili pepper in general, any variety of which descend from the chili peppers cultivated originally in modern day Mexico. In American English we call them chili peppers or hot chili peppers interchangeably, although the "hot" is often added to distinguish them from varieties bell peppers or sweet peppers that were cultivated from the same plants.
Why would this not be i am eating a pepper or i am eating a chili pepper?