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.
Any private key is actually an integer between 0 and 2
256 and is converted to other formats like WIF. (That's for making the copy-pasting easier and decreasing the chance of mistakes with including a checksum).
It should be noted that not any integer between 0 and 2
256 can be a valid private key.
due to secp256k1 ECDSA standard, the number of valid private keys is a bit smaller.