If I were to move schools during the second semester of my junior year, would my rank count as the one of my new school, or my old school ( both public schools)
It would be the school from which you would graduate - your new school. And you would have to ask that school how it would handle your transcript from the old school - not count the grades, give full credit for the grades, or something else.
Some high schools may try to protect their current students’ class ranks by giving unfavorable weighting treatment to inbound transfers’ courses (e.g. giving weighting only to their own honors courses but not those from other high schools). There have been stories about this from Texas, where class rank is the main factor in admission to Texas public universities.
If it’s a transfer within same district, it should be easy but rank would be according to new school. If between different district, new school’s rules apply and they discourage system gamers transferring just to artificially inflate their rank.
The new school. This happened to my son when we moved from California to Texas late in high school. It’s why I feel rank is not indicative of intellect or work ethic, he was top 3% in California school, and moved to an incredibly difficult and highly regarded school, and is taking harder classes and learning more, yet is barely top 8% (with 10 AP classes and 1520 SAT), although he will be much more prepared for college now, despite the lower rank.