-2^2 should indeed be interpreted as -(2^2) = -4

For example, everyone agrees that: 2^2 = 4
If you multiply both sides of the equation by -1, you get: -2^2 = -4

You could argue that it should be written as: - (2^2) = -4
But if you look at polynomials:

5x is a shorthand for 5*x
and -5x for (-5)*x,
so -x is a shorthand for (-1)*x

5x^2 is a shorthand for 5*(x^2),
-5x^2 for (-5)*(x^2)
Then why would -x^2 be anything else than (-1)*(x^2)?

If you don't want to accept the same rules as the ones for polynomials, consider this:

* the order of terms doesn't matter for an addition:
for example:
2 + 3 = 3 + 2 = 5
2 + 2^2 = 2^2 + 2 = 6

* subtraction is just addition of the opposite:
a - b = a + (-b)
for example: 5 - 3 = 5 + (-3) = 2

So, a - b = a + (-b) = -b + a
for example: 5 - 3 = -3 + 5

So 2 - 2^2 = -2^2 + 2
And 2 - 2^2 = 2 - 4 = -2, which means -2^2 + 2 = -2 and -2^2 = -4


Another example,
to solve equations like ax^2 + bx + c = 0, there is this method:
D = b^2 - 4ac
solutions: (-b + sqrt(D)) / (2a) and (-b - sqrt(D)) / (2a)

-x^2 + x + 2 = 0

D = 1^2 - 4*(-1)*2 = 1 + 8 = 9

solutions: (-1 + 3) / (-2) = -1 and (-1 - 3) / (-2) = 2

To check your solutions, you can substitute them in the equation:

-(-1)^2 + (-1) + 2 = 0 ? ... only if -(-1)^2 = -1
-2^2 + 2 + 2 = 0 ? ... only if -2^2 = -4

I think this shows the convention of the unary minus versus powers.