How do I rename my iPhone in the Apple Account Device Menu?

Name can come from different locations, depending on where it is listed

The easiest place to check is

 Setting > General > About

Is that the old name? if so change

Re: "Apple continues to show an iPhone 4 that I have not had in years." Not sure which of the following you are referring to:

Deauthorize device in iTunes: Deauthorize your computer using iTunes - Apple Support

Remove device from account: What to do before selling or giving away your iPhone, iPad, or iPod touch - Apple Support