Agree, they will make the determination from the admission letter. Normally you don't even provide flight or accommodation details for X visa applications. You should be aware that the X1 comes with more time but also with more hassles, compared to the X2. For the X1, once you arrive in China, you will have to undergo a medical exam and there will be a bunch of extra paperwork, since within 30 days of arrival, X1's are converted to Residence Permits (X2's are not and do not require the medical exam). With an X2, you basically just show up in China and that's it, unless you are planning to leave China and return during your visa period and your X2 was single-entry.
Chinese public and private schools/universities end their first term a couple of weeks before the start of Chinese New Year, and all visa officials will know this. You should just go ahead and ask for the time you want (as long as it's under 180 days) even if it runs past the school term. Then see what you get. If you want to spend a lot more time in China than the X2 allows, you may just need to go to Hong Kong and get a new L visa and come back in as a tourist. As long as you're in China on an X, the school is responsible for you and your actions, and few schools want you in China for an unlimited period after the school year, under their watch.