Yes, I do believe you are correct. The way I remember it, there was a full recount that kicked in automatically, by state law. When that recount started going in Bush’s favor, and the margin of victory was widening, the Dems sued in Florida court to stop the full recount. Then they fought in court, all the way up to the USSC to attempt to get them to order only a selective recount, and the Supremes refused to interfere in the process by overturning the state courts. Then, finally the full recount resumed and W was declared the winner.
At least, that is my recollection.
They tried to “cherry pick” and were afraid of the full recount.