Hi.
You may cause a stir with your question - just giving you a heads up. Refillable mug debates are a hot topic!
Here's the simple answer to your question: If you want to be able to get free refills on drinks from BOTH resorts, then yes, you will have to purchase a mug from each resort respectively.
IMO, if you're only going to be at CSR for a couple of days, save your $$$ and buy your mugs at WL where you will be for a longer period of time. If you're counting on those free refills, buy mugs from both resorts. Then you have souvenirs. But I think if you take a CSR mug to the WL and fill it up that you're "not doing the right thing." Of course I have seen people filling up all sorts of mugs, cups, containers, etc. at various resorts, but it doesn't make it right. And that is against Disney policy anyway.
Hope this helps!!!