Correct. The definition of service animals is the same. But for air travel, the ground rules are different because not only service animals are covered.
I don't think miniature horses are allowed on aircraft any longer, although I have seen pictures on the Internet of horses in airplane seats. Photoshop? Maybe.
But the new rule makes it clear -- for airline travel, ONLY dogs can qualify as service animals.
The thing that's different with air travel is that ESAs were also granted flying status, and airlines were not allowed to charge for them.
Now that has been completely changed. After Jan 1, airlines don't have to accept them in passenger space, they can charge fees for them, and they can require that they travel in the cargo hold.