You are not wrong in the way you explained what a private key is because the
Bitcoin Wiki refers to private keys as "
secret numbers" as well. But I have always wondered why they opted for that terminology? Why call a string made up of letters and numbers (alphanumeric string) a number? I guess there is a logical explanation behind it.
A private key is just a random series of ones and zeroes. You can generate it manually by flipping a coin 256 times. You may have heard of elliptic curve cryptography. A private key is essentially a multiplier that shows how many times we should add the base point G (constant point on an elliptic curve) to itself. The result of this multiplication, that is, elliptic curve scalar multiplication, is some point on an elliptic curve also known as a public key. So, both point G and the point of a public key are on a curve while the private key is not. Public keys aren't random numbers because they are determined by a specific multiplier, but that multiplier is generated by you randomly. Only you know this multiplier, only you can sign a message using this number.