The more I think of this, I know the answer and it is, renaming happens during the download, and applying the memorized payee happens later (which may or may not be considered part of the download depending on the download mode).
When I say “download" I mean the process before the user gets to interact with the transactions.
Quicken Windows has two download processes. One where the transactons go directly into the register and the other where they go into the Downloaded Transactions tab at the bottom of the register.
First let me talk about the process with the Downloaded Transactions tab is used.
When you look at the transactions in the Downloaded Transactions tab you will notice the the payees are already renamed. That shows you that process was done during the download.
But if you will notice that the Downloaded Transaction tab doesn't have a category. When you select a transaction in the Downloaded Transactions tab Quicken will show you the transaction in the register. It is at this point that you can tell what category is going to be assigned based on the payee matching the memorized payee. To me that indicates that this is when it done, not during the download. And I might add that matching to an existing transaction in the register is “dynamic”. As in say a transaction in the register is matching an existing transaction in the register. If you change the amount of that transaction in the register, it will unmatch. Same with changing an existing transaction in the register to get it to match.
When not using the Downloaded Transactions tab (Automatic transaction entry mode) what happens is it goes throught same process as above, but adds the extra step of accepting each transaction into the register before it returns control to the user. It again that second part that actually uses the memorized payee to fill in the category.
Note that the automatic entry mode also sets the status to New or New Match so that people can review these new transactions to make sure they are correct. In the case of using the Downloaded Transactions tab this isn't needed/set because the act of accepting a transaction into the register is its “user checked the transaction for accuracy".
BTW note that the memorized payee is used during manual entry, but the renaming rule is strictly used only during the download.