Is it okay that my cell phone still shows the name of the previous wireless service, even after I unlocked my phone?

It is perfectly fine that the name of the previous network service provider appear on your mobile phone, even after you have unlocked. The logo has been programmed into your phone by the original service provider as a splash screen before it was sold to you. This logo has no impact on your unlock service.
Unlocking your phone reverses the electronic barrier put in place by the original service provider. Unlocking your phone does not effect your splash screen and vice versa. The good news is you can get remove the splash screen. See if you can find it in the phone’s manual on how to change the splash screen, or consider calling the mobile phone manufacturer and have them give you step-by step instructions on how to remove the splash screen off the phone.