The definition a prime number is having two factors. As the multiplicative identity, one only has one factor. Some might see primes as being numbers with >=2 factors. This interpretation is less useful. What is completely useless is a definition that makes exclusions of specific numbers as dogma. What does that say about anything? It’s totally arbitrary and unexplanatory.

No, that is not the correct definition of a prime. You can find the right one on Wikipedia.

The reason that 1 is not a prime is to preserve unique factorization. A basic fact from number theory is that every integer uniquely factors as a product of primes, say 21 = 7 times 3. If 1 were a prime, we'd also have 21 = 7 times 3 times 1. That's bad.

In more general number rings, other units (e.g. i) are also not considered primes for the same reason. The exclusion is not arbitrary.

As a complement of your comment: For more information, see Unique Factorization Domains (https://en.wikipedia.org/wiki/Unique_factorization_domain) for the generalization of how prime factorization work on things other than integers.

For a mathematician the suggestion that 1 might be a prime is on a par with a suggestion that a circle is technically a type of triangle.