I think the order is 奶酪> 干酪，乳酪 > 起司，芝士 in Mandarin in Mainland China. As 奶酪、乳酪、干酪 are all Chinese inherent words, while 芝士、起司 have the same or similar pronounciation as 'cheese'. http://ccl.pku.edu.cn:8080/ccl_corpus/

In Taiwan 起士 = 起司 > 乳酪>>>>>> 奶酪>>>>>>>>>>>>>>>>>>> 芝士 = 乾酪 We use 奶酪 as milk pudding here. Barely think as cheese.

奶酪

I think "奶酪" is the most common word for "cheese".

It depends on where are the people from you communicating with. In China mainland, people tend to use "奶酪", "乾酪" and "乳酪". All these 3 words show how Chinese definite it according to its characteristics. It's a example of what we called "意译" (translation according to original meanings or just give a new short definition). "奶酪" is more often to use. In Taiwan, people tend to use "起士", "起司" and "芝士". All these 3 words are sound similar/close to the original word "cheese". It's a example of what we called "音译" (rough transcription according to pronunciation). And what mainland Chinese say before taking a photo is ”茄子“, which is also a rough phonetic transcription of "cheese". I haven't noticed its etymology until you mentioned it. But I don't know which one is more often to use among these 3 words. I can't speak for everyone, but it's like a general case. Maybe there are some differences between all these 6 words, yet many people like me just don't know and don't care about the possible differences. BTW, mainland Chinese have a word "牛奶布丁" for "milk pudding".