Example of Random Binary Tree?
A random binary tree is any tree that is generated by any random process.  So, yes.  You have described a random process that generates a binary tree; you can describe the result as a random binary tree.

At a technical level, if you have any random variable whose values are binary trees, it would be reasonable to call that a "random binary tree".

Wikipedia is saying that there are two random processes that are particularly commonly considered.  But those aren't the only possibilities.  You've described a third possibility, and that's also valid.