Now that I received my GC through employment, does my employer need to change my position to the one filed in the Labor Certification?
Yes, now that the GC is approved, your employer should "permanently" give you the job described (including the title, salary and job duties) in the Labor Certification. This change should take place within a "reasonable time" after the GC approval.
Keep in mind, "permanent" does not mean forever. This term describes a job that has no pre-decided termination date. We see no violation of the law, If the employer, due to economic or other circumstances, can no longer support the job after having offered it to you in good faith.
As to what is a "reasonable time," we would have to look at the circumstances of each case.