A few weeks ago I wanted to try the new Pog flavored Sprite at the Poly Tower, so I bought a cup before our Wailulu ADR. I did take that into the restaurant with me since I paid for it, and I had 2 or 3 more refills within an hour. I felt kinda bad about it, but then I told myself that if I didn't have that drink, I would just be ordering water that was free. I did buy my DD22 a celebratory cockail (she had an awesome 1st year in her doctorate program) which cost me $17 or $18, and if I didn't buy that, she would have just had a water too. In the end, our server got a way better tip with me having my soda from outside the door and my DD's cocktail, then if we would have just ordered waters.
I can see how bringing in a refillable mug, might take away from the servers final total that affects the tip, but then again, I would think that servers working in a WDW resort restaurant would somewhat expect people to bring in their refillable mugs - especially kids. Those mugs aren't cheap, and people are going to want to get the most use out of it as they can. I personally don't buy one, but my DD does and she fills that thing every chance she can get, which is what it is intended for.
Someone mentioned about brining a Yeti into a restaurant, and I have done that too. If I'm just drinking water, and I have my Yeti with me, then I'm bringing it in, because sometimes I don't like the taste of restaurant water. When my DD was still playing softball, and it was a 95 degree day and I'm drinking Gatorade or something like that, then the Yeti is coming in with me. Nobody has ever told me that it wasn't allowed. In fact....my DD and I both had our Owala with us this past trip, and we had that everywhere with us at WDW, and nobody ever said we couldn't have it.
In the end, I'm a good tipper, and I bring a lot of cash with me on our WDW trips, so I can pay all tips in cash. So our servers aren't being neglected because we bring in a refillable mug or another drink container. Many times I use the final total as just a starting point for my tips, and if we get good service, than I will go more than 20%, because they deserve it.