I really am begging people to stop trying to come up with clear-cut definitions for bi, pan, and omni that attempt to make out differences between them.
Both in bi history and in present bi people, there are definitions of "attraction regardless of gender," "attraction to two or more genders," "attraction to all genders, varying depending on gender," etc. All of those are valid definitions of bisexuality, and to claim otherwise, to try forcing it into one definition, is both bi-erasure and biphobic.
And I'm not saying that the lack of a possible, clean definition is a problem. It's not. People can choose pan as a label because they experience attraction regardless of gender and want to make that clear. People can choose omni because they experience attraction to all genders, but it varies depending on gender. Hell, you can choose one over the other because you like the flag better or the word or it resonates more with you. You can choose all of them if you want to.
The thing is, though - there will most likely be pan people who experience attraction differently, and still choose pan. They're still valid. Same goes for omni, and bi, etc.
When you look at this historically, it happened something like this: queer history has always been erased wherever possible. A whole generation of us has been erased. This happened too, and in some regards even more so, with bi history. There are countless sources from as early as the 70's that define bisexuality as attraction where gender doesn't matter at all, as attraction to one's own and all other genders, and just about any definition you find floating around the internet for bi, or pan, or omni today.
A lot of this history and these accounts were forgotten. As a consequence, in the early 2000's and the rise of internet queer culture, people assumed "bi = 2" and came up with pan. Now, I want to make it very clear that I'm not blaming anyone for this. There were thoughts along the lines of "this excludes trans people, and we want a label that doesn't do this."
This is both transphobic because trans women are women, and trans men are men - not some other gender that requires a whole different form of attraction - and, again, bi erasure. Historically, the bi and trans communities have always been closely interwined. In fact, many people are trans and bi. Bisexuality has also, in many spaces, as early as the 70's and 80's, disregarded the gender binary and rejected it. Bisexuality was not and is not about binaries, and the experiences and lived realities of it always had the whole point of refusing a clean-cut definition.
And again, this does not mean, as some people like to claim, that we should disregard other labels, or that bi is the 'umbrella term' under which labels like pan or omni fall, as some kind of sub-category. They can co-exist perfectly fine, and people can pick whatever label they're happiest with. Because labels aren't there so that we can be put into neatly labelled boxes. They're there so we can understand ourselves a little better, or because they make us happy, or help us describe lived realities. There's no sense in pitching us against each other, by ourselves.
But this also means that people got to stop erasing us, our history, and our lived realities. Stop putting us into a box so you can neatly close your own.
If you want some more info/sources on this, this article is a good place to start.