Yes, it is perfectly okay to transition in any way that you want. It does not make you selfish. You don't "owe it to your culture" to continue repressing by pretending to be something that you're not. There is nothing wrong with being trans, and there is nothing wrong with you being yourself.
You're right. Maybe there's a reason "going against your culture and parents to do the right thing" is a common trope, and we consider the charactes who did it hero instead of villain.
Still, it'd be hard to reprogram decades of anti-trans stance brought on from my culture, even more so when it's still proliferating and as things gets worse for us.
As for tackling your internalized transphobia, I would encourage you to stop intentionally seeking out transphobic content online - as you have already identified it is a form of digital self-harm.
It's easy for me to avoid them most of the time, but it's very hard when my mood is low. Worse, most of my transphobia nowadays comes from my head, not contents from the internet. I can't exactly turn off my brain to avoid thinking of transphobic thoughts...
block yourself from going there if the impulse is too strong.
A good number of transphobia I had is from anti-trans laws news, which I find easily from most trans communities. When I hear transphobic laws passed, I tried my best to justify them even though I didn't agree with them. It's like I'm putting effort to become transphobic, even though deep down I don't want to.
I would recommend talking with an LGBT+ friendly therapist if you are in a position to see one.
I think I might if things get worse, but there's practically no LGBT friendly therapists around, nor do I have the money to see one.
Maybe I can contact a trans friendly hotline if they operate worldwide. Maybe like the trevor project or something like that.
The story of how I got my name was both interesting and mundane.
I played a videogame named Stardew Valley and created a female character there. I don't really think about it much when choosing her name, I just wanted a plant-related feminine name so I chose Jasmine.
Many weeks later, my egg cracked and I randomly chose Jasmine as my new name. Later that day, I realized that the name was the same as my Stardew character.