Why is n^2 considered a type of time complexity but n/2 isn't? [duplicate]

Question

I have searched all over stack overflow and couldn't find a concise answer for this. Consider this code:

n = [ [1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12] ] # O(1)
for i in n: # O(n)
    for j in i: # O(n)
        print(j)

If my understanding of O() is correct, the time complexity for this algorithm is 1+n*n, and if we throw away the insignificant 1+, it becomes n^2, or quadratic time. Now consider this code:

n = input() # O(1)
for i in range(0, len(n), 2): # O(n/2)?
    print(n[i])

This time we use division instead of an exponent: 1+n/2 -> n/2.

But everywhere i searched, people said O(n/2) == O(n), because O() measures relative time, etc. But if O(n/2) is absolute, and we should collapse it into O(n) when using big O, doesn't that make types of time complexity like log n and n! incorrect too, because it would just mean modifying n with absolute values...?

n^2 == n because it's the same as n/2, but instead of / we use ^. n! isn't correct because factorial is just a series of multiplication, so n! is the same as n*(n-1)*... which, i believe, would collapse into something completely different after throwing away all the -1s. Where am i wrong?

P.S. My understanding of logarithms and big O notation is very incomplete, please don't hate me if i'm missing something obvious.