Perhaps I am reading it wrong.

Wolfram states that a prime number (or prime integer, often simply called a "prime" for short) is a positive integer

More concisely, a prime number

Wikipedia states that a prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Are these examples generalizations, or is that what people really think?

These definitions are far too restrictive in the sense of what a prime really is. they are far simpler entities than that. However they can be represented neatly in the contexts that are given.

integers, Reals, naturals, Primes, ect.. Could someone give a definition that would describe what is accepted as a standard.

and if a definition exists today that is to restrictive, how would one go about generalizing it.