To add to the other answers, Smaug in the Hobbit has four legs and two wings. Not two legs. That's just an invention of the movie makers. Tolkein makes that clear in this sketch:
(death of Smaug)
So Smaug at least was a dragon, not a wyvern in the original book.
Which support's Laura Mitchell's answer.
See also
Also a dragon from an older story: Glaurung sets forth to seek Turin.
110 Drawings and Paintings by J.R.R. Tolkien: Of Middle-Earth and Beyond
Also his sketch of a baby dragon
Dragons - Tolkien Gateway
All these paintings are by the author himself, so can't doubt that he thought of his dragons as having four legs in addition to wings.
For a modern example, the dragons in "How to train your Dragon" and its sequels and long running TV children's series on CBBC are a mixture of wyverns and dragons.
For instance Hiccup's black dragon Toothless, in the center has four legs while Astrid's blue dragon Stormfly (changes colour depending on its mood) has two legs so is a wyvern.
This is the CBBC website about them.
Dragons - Defenders of Berk
This short youtube featurette introduces some of the dragons, so you can see that some are wyverns and some are dragons
Page on youtube.com
So Smaug at least was a dragon, not a wyvern in the original book.
Which support's Laura Mitchell's answer.
See also
110 Drawings and Paintings by J.R.R. Tolkien: Of Middle-Earth and Beyond
Also his sketch of a baby dragon
All these paintings are by the author himself, so can't doubt that he thought of his dragons as having four legs in addition to wings.
For a modern example, the dragons in "How to train your Dragon" and its sequels and long running TV children's series on CBBC are a mixture of wyverns and dragons.
This is the CBBC website about them.
Dragons - Defenders of Berk
This short youtube featurette introduces some of the dragons, so you can see that some are wyverns and some are dragons
Page on youtube.com
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Robert Walker
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