Does H-1B approval also mean COS to H-1 ?

[social.parag]

As per my employer's law firm website, an email confirmation of H-1B was sent on Oct 4, and they also received H-1B approval notice on Oct 11. However, from SEVIS, I can also check that my current status is still J-1 (as of Oct 11). I was under the impression that if H-1B petition is approved, it also means status is changed to H-1. Are they two different things?

(I did have a waiver for the J-1 home residency rule).

I am not able to get this information from the company's law firm, as they simply do not answer my emails or phone calls.