Both 〜と呼ぶ and 〜という can be translated as “to call...”.
So, for situations where something is given a name, what's the difference between the two?
～と呼ぶ is used to tell someone to call you by that. For example. Joseph Joster from Jojo no kimyouna na bouken famous phrase is:
Which translates to "Joster, my name is Joseph Joster, call(address) me by Jojo" It's used to tell the others to call you by something you specify.
My name is Momotaro, call me by "Momo".
Nice to meet you, I'm being called Aria, call me "Ari".
This example has both the "iimasu" form, which is used to introduce yourself, and the "yobe" to tell people to call you by "Ari" name. "言う" is used for introduction and it's not formal either, don't forget to use the masu form with it, while "呼べ" is used to tell people to address you by something. That's the use of "呼べ" in your question and not in general. It's meaning is "to call".
For example. モモを呼べ means "Call Momo!". Don't forget that. I hope this helps.