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This doesn't make a lot of sense to me, tbh.
I am all for recognizing that languages change all the time, etc. But also, isn't the point of laying out categorical schemes like this to provide clarity? And I think clarity calls for us to, when possible, consider etymology, history of usage, etc. because that allows people to more readily understand terminology when they just encounter it in use, rather than getting it from some kinda chart like this.
So, I think it's weird to call bi an umbrella term if the term and the terms supposedly falling under the umbrella are exclusionary. Example: "communist" is an umbrella term. You can call both Marxists and Ancoms communists, because they both fall under the umbrella. I agree that you shouldn't call a pan person bi, but that means pan doesn't fall under any kind of bi umbrella.
Also, pan and omni are synonymous; pan being Greek and omni Latin. I've never heard of this distinction of pan having no gender preference and omni having one - feels made up by this person. BTW, I identify as pan, and have a preference (though I personally wouldn't say it's a gender preference. I know several agender people and I am attracted to some, and not to others, though none of them have a gender, so I don't think gender is really what my preferences are about. And I could go on about this, but that's a whole nother post).