Imagine someone gave you the following expression: 2\*(2\*4x\^2)+5. Is the first term's coefficient 2 or is it 16 (because 2\*(2\*4x\^2) simplifies to 16x\^2)? Or is it reasonably to say any factor of 16 is a coefficient of the first term?
As a rule of thumb, the answer to any question you might have about a mathematical object/expression shouldn't change depending on how the object is written down. It should always be 16.
The expression is a polynomial so the coefficient of the first term is 16.
The first term is all multiplication so you have 2 * 2 * 4 * x * x. What does that equal?

It doesn’t matter what order they are written in, what is x* 2 * 4 * x * 2?
The coefficient of a variable term is the number the variable part is multiplied by.

Not some selection of factors of that number.

You ask for sources, but I expect you will find this definition in any introductory algebra text.

You don't have to simplify it. In your example, you could say the coefficient is 2 * 2 * 4 or 16, since those are the same thing.
> 1. a numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g. 4 in 4x^(y)).

This is Google’s definition of the word, so it’s not necessarily the most accurate thing in the world mathematically, but it is definitely *at least* the level of understanding you need to have. In 2*8x^(2), the 2 cannot be a coefficient because 8x^(2) is not the kind of object that can have a coefficient. The coefficient is the number (16) next to the variable (x^(2)).

Also, if you’re trying to think about what it means, it probably doesn’t help to say “the coefficient of the first term”. 4x^2 + 6x and 6x + 4x^2 are obviously the same so is “the coefficient of the first term” 4 or 6? You can instead say that 4 is the coefficient of x^(2), etc.