I came out on the diversity portion and I am having deep regrets and fears that anybody outside the admissions committee sees it. I doubt anybody on admissions even cares but I need to know that such information will not spread.
I shared this information on a portal that asked for a diversity statement and character traits about me as part of "holistic review", not on the statement of purpose which I assume is for professional goals.
We are missing a lot of detail from the question, but:
Then the usual cultural expectation is that someone's LGBTI identity is not shared without their consent. Occasionally people do things that are unexpected.
In the current climate, at the average university people might be surprised that you have not been out for a long time. They will not be surprised if you are experiencing discrimination outside the university.
There are certainly universities that are not friendly to LGBTI students in the USA, but most of them do not have doctoral programs. Usually they can be identified by reading their websites, including mission and diversity statements, carefully.
As a fairly substantial long-term data point(s)... from my being involved-in and observing grad admissions at a big state univ:
At my univ, the central admin people hardly look at applications, to begin with. Documents and info are entered into the system by a certain few people, but those are just data-entry people, and I'd be surprised if they looked at things more than for formatting and completion of application.
At the departmental level, especially for the last 10+ years, all applications are electronic, and the only people with access are the admissions committee people, who are tenured (or occasionally tenure-track) faculty. Each person has roughly 40-50 files to look at, at least, so there's typically no specific connection of names with data, after recommendations have been made.
It's not so much that there's a well-known rule about sharing info, though there probably is such a rule, but just that no one finds any of the info in this ocean of applications interesting enough to gossip about, so, in fact, there is no sharing.
In my univ, the grad admissions cte does pay some attention to the diversity statement, in the sense of hoping that people express concern and interest in such issues, whether or not they themselves have had hardships, etc, related to such matters. Yes, direct personal experience is worth a lot, of course, as with most things. "All other things the same", we'd prefer to admit grad students who understand that "mathematics" is a human activity, and that equitable behavior and appreciation of varying backgrounds and privilege is important, for many reasons.
(No, not everyone agrees with my last sentence...)
