Okay, two of the coffee cups recommended to me do NOT fit into a standard bicycle waterbottle cage (at least the two models I use on my bike). These were the OXO-Good Grips stainless coffee cup and the Contigo Extreme mug. Both were too big to fit easily into the waterbottle cage.
The Thermos Raya Stainless Steel Tumbler kept my beverage extremely hot for over an hour in my freezer. It fit very snugly into my waterbottle cage, not rattling at all. HOWEVER, the lid is merely pushed into the cup portion. When a mildly warm or cold beverage is in the cup there is no problem. When I had a hot beverage in the cup, the lid was loose from the expansion of the cup (due to the heat) and easily fell out when I turned the cup upside down and gave it a mild shake. I would hate to see what would happen if I was sipping and hit a pothole.
I did find a OXO Plastic Liquid-seal Travel Mug. It was slimmer in profile than the stainless steel model and fit my bottlecage perfectly. It did not keep the beverage hot as long as the Raya, but the lid is a screw-on type lid and the push button feature worked perfectly.
All in all, the OXO Plastic Liquid-seal Travel Mug is perfect for my needs. Looking online, the stainless steel model should be the same size. I'm wondering if the model I tried was a larger variant. I'll look for a stainless steel model that's comparable. Till then, I am retiring the Raya and switching to the OXO model for my commuter mug of choice.