Actually, the operator name is stored in the simcard and also the mobile phone vendors upload these network names to the phones. Some of the simcards capacity are very small like 16k capacity so I am not sure whether it can carry all these operator names and codes. If the MCC+MNC is different from the HPLMN then the network name saved in the mobile phone will be displayed if the name is not stored in the mobile phone you will see the MCC+MNC on the mobile screen.
In roaming scenario, you will see the HPLMN and VPLMN network names changing to and fro but it will finally reside on the VPLMN network names.