I used to use a different Farm Innovators heated bird bath, but it was so shallow that I had to keep filling it up daily. ( https://www.amazon.com/dp/B00596UOZ6/
Then I got one of their rock-type heaters to put in my existing bird bath, and that is working out much better. There was no warning though about not using it with any extension cord.
I do use a 10' extension cord with it, a good heavy outdoor rated one. The top connection is closer to the eave so it's pretty protected there. At the lower end that plugs directly into the rock heater, I use this cord housing. ( https://www.amazon.com/dp/B002MHFXS8/
I've used it this way for several years now without any problems at all being outdoors with either the heated shallow birdbath or the rock-type heater. Nothing lays in the snow though ever, the extension cord runs right next to the house and hangs high enough to never come close to touching the snow or the ground. If you had one continuous run of a sturdy outdoor-rated extension cord and protected the connections, I wouldn't think it would be a problem. Probably be expensive for that long of a cord though.
Edit: Just to add that every heated birdbath item I've ever seen comes with a very short cord attached, like 12"-15" max. There's no way you can hook them up without using an extension cord. Unless there's something very odd about that particular bowl that you saw, I really don't see any problems as long as you use proper 3-prong cords, make sure the wide blade is in the wide slot, etc... just the basic common sense things you would normally do.