@decaffeinatedghostkid since you asked nicely, I will!
So first of all, I have a deep, deep dislike for that definition of lesbian, for multiple reasons. I understand it emerged as an attempt to be inclusive to nonbinary lesbians, but as a nonbinaru lesbian, it actively hurts and excludes me from lesbianism.
For one, as an agender person, I dislike having all gender identities that are not stricly "man" be grouped under one label. "Non-man loving non-man" could apply to two agender people in a relationship, but they may not want anything to do with lesbianism. And lesbianism is still connected love for women/womanhood; by broadening the definition this much, "lesbian" essentially becomes useless for anything but excluding anyone who may be defined as a "man". It centers lesbianism around what it isn't instead of what it is (something big and expansive and rich with complexity).
Second, it is a definition made without multigender people in mind. A person who is both a man and a woman might identify as a lesbian*, and it's exorsexist to tell multigender people that we can't be a certain sexuality because monogender (often cis, not always) people don't understand how our gender works, and demand that our identities and labels conform to monogender understandings of the world.
The definition I, and many others, prefer is "someone with queer attraction towards women". But I think it's important to know there will never be a perfect, all-encompassing definition for lesbianism. The denotative meaning for most people is a woman who is attracted to women, but people may identify as lesbians for all sorts of connotative meanings. Because lesbianism has a lot of connotations, and even if people aren't technically "women who are attracted to women", they have a more personal, specific connection to lesbianism that leads them to use that label.
A trans man, for example (and to actually answer your question), might be someone who has exclusive attraction to women, but it doesn't feel straight. Maybe he spent a long time in lesbian spaces before realizing he was a man, and even if he feels entirely like a man, his attraction to women does not feel heterosexual. So, he identifies as a male lesbian.
This is very much not a new phenomenon. Trans men have been in lesbian spaces, in general forever, but as lesbian trans men specifically for decades. It's not uncommon to read/hear things about FTM lesbians, who stay in their communities and even with their cis lesbian partners after transitioning, while identifying as lesbians. In the book "Butch Queens Up in Pumps", which talks about the gender system of drag balls, "Butch" is the gender label for masculine/GNC women as well as trans men; both groups identify with the same gender label in that system. Trans men have a long history of identifying as lesbians even after coming out and even medically transitioning.
Or, we could go back to the man-woman lesbian*, who might call themselves a straight lesbian, because they are only attracted to women but are both a man and a woman, therefore both a lesbian and heterosexual.
Queer language is not a taxonomy, crucially. We are not trying to scientifically classify people's sexuality or gender. If we were, then we would need consistent, clear definitions of things like "lesbian"; but that's not the goal of the word "lesbian". We use it to find solidarity, shared experience, community, safety, acceptance. There is no Queer Scientific Society who produces the definition of words and sorts people into different groups, we just form communities and then opt in or opt out depending on how we feel the most comfortable.
And when it comes to trans people, things with sexuality get even more complicated. Binary trans people often do not have the exact same experiences as cis binary people, so they may not identify the way a cis binary person might (where a cis man might identify as straight, a trans man might identify as lesbian). And nonbinary people, such as girlboys/androgynes/similar folk, have "conflicting" genders. If we can't identify as lesbians despite being women because we are "tainted" by also being men, and if we can't identify as gay despite being men because we are "tainted" by being women, we end up being forced out of sexualities for no other reason than monogender people don't want to question their binarism, even if those labels are the ones which make us the most comfortable, represent our feelings the best, or if they are the communities we have been in for a long time and still feel connected to.
