Are these buses marked or numbered or identified in some way that someone on Beach Road can jump on one and head up South Pattaya Road rather than having to get off a bus that turns onto 2nd Road and get on another one?

No. Pot luck.

Route numbers on buses are pretty much meaningless. They go where they like anyway.

 

Twice I've had one that continued to Jomtien, which is where I was going. Another time, though, other passengers wanted to get out by the school and all other passengers were then forced to get off as well and get on the bus in the queue.

Happened to me when I was going from BR to Jomtien. Got on a baht bus driving down Beach Road. At the junction at the school, all passengers were thrown out of the bus. Then the exact same bus and same driver was then lining up at the queue to go to Jomtien. So we were thrown out of the junction just so that everyone would have to pay him twice.

Are these buses marked or numbered or identified in some way that someone on Beach Road can jump on one and head up South Pattaya Road rather than having to get off a bus that turns onto 2nd Road and get on another one?

 

I did get on a bus marked 6 near Soi 10, Beach Road. It did take me along South Pattaya Road. I got off near 3rd Road but I did see the bus continuing along South Pattaya Road on it's proper route. That said, many do not. They turn off as you say at 2nd Rd or Soi Buakaw.

 

Try for one marked 6, that is your best bet but no guarantees .