Would a person have had to be a child prodigy to be called a genius?

That really depends what 'genius ' means to you. There are exceptional people who changed lives because they were able to put into use what they know regardless if by record or tests they were outstanding or not. And there are geniuses who are good at tests, best at memory and are exceptionally outstanding when it comes to thinking, problem solving.

Bill Gates for example, was a college drop out, Zuckerberg is, and they move the world being the richest men, and being ahead of their game in communications and technology. There are men far more intellectually superior than them.

Here's what's important, life is not a race. One flower does not compete with another flower, it just stands in it's place and opens in full bloom on it's own. You have to know what you're good at stop hustling and competing with other people. Be more excited about life and less fearful.

My understanding is that a prodigy has extraordinary talent and skills beyond his/her age in only one area. The prodigy might be utterly ordinary or even lag in everything else. So you get a kid who can play the harp like a professional at age 8, or design machines like an engineer by 12, but they may read at or below grade level and have trouble with simple math.

The genius demonstrates from an early age a comprehension, problem-solving, and original thought in whatever his or her interests are, across the board. While they have strengths and weaknesses, they're generally "off the charts" in most or all their academic and artistic endeavors--but possibly not at the level of the prodigy in his or her single area.