Not an expert, but my understanding is that antique = old,
patina = the worn - in look from use, wear and age, and
Burnished = rubbed with steel. Usually seems to apply to toe and heel caps.
Any of them can be forced through a number of methods.
For example, a patina is often forced on carbon steel in order to protect against rust. Most often through a dip in something acidic such as vinegar, mustard or lemon juice.
On new leather, antique is usually painted on.
I've read that AE "burnishes" their various shades of brown by running the toes and heels on the black buffing wheel for a bit
My first thought when I saw upr_crusts gorgeous boots was that they were burnished.