Doesn't the name imply only two?

But is there a rule that says a dog can’t play basketball?

As someone who identifies as pansexual, the distinction i make is that I don't consider gender when vetting a potential partner, it's just not a metric that is part of that decision.

Aside from that, your description here is spot on from what I understand at least. Of course gender and sexuality is a social construct so other people may feel differently

Err shouldn't it be the other way around? Ie bisexuality under the umbrella of pan-?

Bi-sexual. So attracted to both sexes. Not genders.

My wife generally says that she's bi, but if she were coming out today she's said that she'd probably call herself pan, that just wasn't really a common term when back when she did come out so she's spent most of her life calling herself bi, and she just kind of identifies with that label more at this point.

To her, the term bi does kind of imply that someone is attracted only to males or females and would tend to exclude non-binary gender identities (though not necessarily trans people, being MtF or FtM does still kind of line up with a traditional gender binary, a trans man is a man and a trans woman is a woman)

Which doesn't really describe her, she'd be cool with any identity, which to her is more pan, but again she's just been calling herself bi for so long it just feels weird to change that, and since she's off the market at this point it's a little bit of a moot point anyway since she's not trying to get in anyone's pants but mine.

I feel like placing identities into hierarchies always ends up controversial. Like non-binary is technically under transgender because transgender just means your transitioning from your assigned gender to something else; but a lot of non-binary people really don't like being called transgender instead of non-binary.

At the end of the day, the words meanings are a subjective experience that cannot be directly observed or compared. So functionally all self-identifying terms exist flat relative to each other because there will always be contradictory definitions that you can't rigidly settle on without ignoring significant groups of people.