There is no reason to select "Begin", the printer is chosen from a dropdown below that section.
Reloading a printer does nothing to the Client PC drivers. The 2 things are not related.
The name at the machine is different when sending RAW vs LPR , that is most likely the reason behind the names changing.
At this point there is no reason to change anything on the print server, you go to the server and add an entirely new printer with the new driver on a new port and set its preferences and defaults and port as specified and test from the server. If successful, share the new one out and test from 1 client, if successful convert the old to match the new and you are done. Nobody is suggesting you just randomly change an in-use shared printer.