I just wanted it to update my case a little bit for anybody who's also going through this.
One month after my injury, I went to see a PT and after some test, he didn't think it's a tear but more like a pinch. I chose to trust him more because 1. I'm going to do PT anyways even it's a SLAP tear, 2. the doctor I went to see before didn't even do any exam on my shoulder but sent me directly to the MRI and when I asked him to write a prescription for my PT(it's required in Texas), he wrote tendinitis.
I have been doing PT for 6 weeks, twice/week and has three more visits left. I made some progress on my shoulder. I already started to climb and I can climb most of the routes as before my injury (I'm not a strong climber anyways). Only have some sharp pain when doing dead hang or overhead reachy moves.
Having trouble doing overhead reachy moves is a very good indicator of a shoulder labral tear. Funny, how many docs send you off to get an MRI before they even give you a full range of motion studies. That's a good way of finding out what's actually wrong. MRI = Day Dreaming. Pseudo-science. Black Art.