With any algorithm complexity analysis you have to define what the inputs are considered to be. For cryptography, algorithms are designed to be constant-time with respect to the non-secret inputs. The secret inputs (which you are trying to protect) usually do not vary from one call to the next (eg, long-term private keys etc) - so can be assumed to be constant.

So while the terminology seems confusing, it’s not actually different. It’s just a different choice of “input” compared to typical algorithm analysis.