API 521 provides guidance in how to evaluate potential causes of overpressure, and how to evaluate the resulting risks.
API 520 provides guidance related to sizing and installation.
As I mentioned above, ASME Sec VIII only provides high-level requirements. It intentionally leaves all the risk evaluation and sizing decisions to the user. API fills that gap by providing practical advice to the end-user, including a procedure for sizing PSVs (API 520 Pt 1) that are compliant with ASME Sec VIII. API and ASME do not offer different methods for the final sizing of PSVs. Possibly you're confusing the API preliminary sizing procedure. That is simply a way to get a preliminary estimate of the PSV size. But, the final sizing must be done using the ASME orifice size and coefficient - API 520 Pt 2 clearly states that. So, understand that API and ASME do not have separate methodologies for the final sizing of a valve.
When a vendor sizes a valve for you, they are doing a final sizing calculation. That is, they're using their actual orifice size and their certified flow coefficient. When you size a valve using the API orifice sizes and API coefficient, you're simply doing a preliminary estimate of the valve size. That's a generic estimate that is not based on any vendor's specific size for that orifice, nor their specific coefficient. Obviously, once you go to a particular vendor, they're not going to use, for example, the generic API orifice size a G orifice. They'll use their specific size for that G oriffice and their certified flow coefficient. When you follow the API sizing procedure to its conclusion, you identify a specifc vendor and update your preliminary calculations by replacing the API generic values (orifice size and coefficient) with the specific valves for that particular vendor. The vendor's actual orifice size and actual coefficient values are refered to as ASME values, because they're certified according to ASME testing & certification protocol.
SUMMARY:
API Sizing = (1)estimate size using generic API values for orifice size and coeff, then (2)update the step-1 calc's by substititing the vendors values ("ASME values" for orifice size and coeff.) for the generic values used in step-1.
ASME Sizing = perform calcs using ASME orifice size & coeff -- this is exactly the same as the last part (step-2) in the API Sizing.