Let's put it differently: yes, it matters, and do the reverse of what you think to save yourself a lot of future problems, it will be clearer in 3-4 paragraphs.
You are a student and you are trying to "engage a conversation" with a professor. Said professor should be interested in your work and should engage because your questions. If they do not engage, it simply mean they had no time, you have no political power, so your email ends up in the pile "interesting, to be answered in a couple of weeks".
Unfortunately, a couple of weeks = never, life is hard, don't take it personally and move on.
However, the professor H may notice that you are writing from the well known universitY and the professor may have an interest in getting their power extended to said universitY. You may even be so (un)lucky your work at the universitY is appearing on some internet pages captured by google as done with the famous professor G at universitY. Said professor H may do a quick Google search and find that. You will be very relevant, because you may be an extremely useful pawn.
Professor H may then became your PhD advisor because of some machination. They will then discover you did not pay the huge fees at universitY, you are not part of the alumni of universitY and you have no good contact with famous professor G. They will be an awful supervisor. No way that such a big ego (there are many in academia) can be a good advisor.
The issue is not really if you should/will use the email from universitY, the issue is what kind of professor is giving "more points" to a student because the student is writing from the email from universitY instead than from X.
Ps: on a brighter note, locally it may be helpful to arrange a visit to a good dentist or to arrange a room to rent (or even a job) to use the email from universitY. Ethical? well, you are doing something at universitY, the issue is the receiving person and the bias they have towards people from the universitY instead of using their mail .mx ...