Auto Insurance: Whose Insurance Company Should I Go Through, My Own or At-Fault Driver's

My question involves insurance law for the state of: California

My car is currently in a shop awaiting an adjuster from the other person's insurance
company, Allstate, to assess damages. The collision report said other person
at fault, and so did the insurance companies.

Now I'm reading that you get treated better if you go through your own insurance company, that
they will be less likely to declare your car a total loss, or give you a bad deal -
especially knowing they will get reimbursed.

I thought it was supposed to be better to go through the other party because even if you're not
at fault, your insurance rates can still go up just because you filed a claim and then made them
do work. Especially if in a couple years, you have to file a claim again, even if you're not at
fault the second time either.

Just wondering what other people would choose?

Oh, the last thing is that I read the collision report and part of the way they determined
fault was by damage to cars. In case there's ever an argument, I thought
it might better if the adjuster who takes photographs before car is repaired be from
person's own insurance company, since they'd probably be glad to share photos and
information just in case. (yes I took photos myself, but these would be more official).

Auto Insurance: Whose Insurance Company Should I Go Through, My Own or At-Fault Driver's