The first command routes the port 19999 on the remote host to port 22 on the computer you're using.
ssh part on the second command connects to 19999 (and thus 22 on your computer) and runs the pbcopy command in the current shell (which is the pipe from command_that_prints_stuff)
Depends on the situation, if you want to run a quick script then yours would be better. If you're spending a lot of time in remote_host then mine might make more sense.
The first command routes the port 19999 on the remote host to port 22 on the computer you're using.
ssh part on the second command connects to 19999 (and thus 22 on your computer) and runs the pbcopy command in the current shell (which is the pipe from command_that_prints_stuff)