You are right.
H1 and L1 do not make any difference in GC process, so H1 and L1
is interchangeable in my last post.
Summarized this,, there are 3 options if H1/L1 spouse is already in the US.
1. Let H1/L1 spouse go for GC process by his/her own based on employment.
Sponsor company needs to agree to support GC process in this case.
2. Wait until GC husband/wife becomes US citizen and give H1/L1 spouse GC.
3. GC husband/wife files I-130 for H1/L1 spouse.
From processing speed point, 3 is probably slowest(5+year). Which of 1 or 2 is faster
depends on this couple. If GC husband/wife already spends a few years on GC, 2 might be fastest.
In either of 3 options, H1/L1 has to maintain legal status in the US
on his/her own, independent on GC husband/wife.