They obviously can calculate position, but it's not required for GSM handover. The BTS (cell tower) will broadcast the connection info about neighbouring tower

They obviously can calculate position, but it's not required for GSM handover. The BTS (cell tower) will broadcast the connection info about neighbouring towers and your device will report back the signal strength for each. The network can then make a decision if it needs to move you. This is known as Mobile Assisted Hand Over. The aim of the game here is maximising signal quality, which is not necessary the same as minimising position to a tower.