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No but it has a bearing on it.

Sometimes. Not always.

If a trans woman is a woman that was born in a male body, and said trans woman finds herself falling for another woman (trans or otherwise), then surely she is a lesbian, as opposed to if she identified with her body which would make for a heterosexual relationship.

She would be lesbian, bisexual, biromantic, pansexual, pansexual, panromantic, omnisexual, omniromantic, or any number of other potential sexual or romantic orientations, none of which have any bearing on her identifying with a gender which she was not assigned at birth. Some trans people find their sexual and/or romantic orientations change after they transition, and some don't.

Jenner complained about wOkE tEAcHeRs turning kids trans. The person I replied to said tongue-in-cheek that their teachers affected their sexuality. To me, this implied that they considered being transgender a sexual orientation. I was simply correcting them.

Or maybe we just throw out the whole idea of orientation altogether, (a) because stuff like the above is just too confusing, especially when extrapolated from this simple case into more complex identities and attractions

This is a very "I don't see color" sort of statement. Lumping all queer people together due to being too lazy to work to understand the nuances of different gender, sexual, romantic, and relationship minorities and the challenges each group faces is akin to ignoring the plights of members of minorities races because it's easier to lump everyone together in a "We're all part of the HUMAN race" sort of way.

and (b) nobody cares anyway now that all orientations are considered equal.

I assure you, we queer people very much do care, and all orientations are most certainly not considered equal. Maybe you are fortunate enough to live in a state/province/local authority or country where discrimination against people for their gender identity or sexual/romantic orientations is forbidden by law, or maybe you are just wholly ignorant due to being a cishet person and can't be bothered to learn about discrimination when it doesn't affect you. Whatever the reason, you are grossly misinformed, and I do hope you are uncomfortable enough that you choose not to remain that way.

There are other models of sexuallity. Things like lesbian/gay/straight/bi etc. or hetero/homosexuality is one based on the relative attraction between sexes or genders... But one could also use the model that simply refers to the subject of an individual's attraction. Gynosexual , Androsexual, Skoliosexual or Pansexual. It's arguably a more neutral way to classify because it draws no particular difference between same-sex or opposite sex attraction. It also tends to work better when dealing with non-binary people because self classification gets weird when you need to use a binary classification system for a non-binary experience.

while this is true, the person who made the post above (the original post in the image) is an idiot and wouldn't comprehend this at all. Because they'd be stuck on step 1.