Is a Farmer villager required to breed?

Short answer: No, you don't need a Farmer. But, for two villagers, you need at least 6 doors (that lead to a covered area).

Long answer:

The key to finding the answer is found within your own question. "Additionally, ..." implies more information came before your quoted part of the Wiki.

From that same linked wiki: "Villagers will mate depending on the number of valid doors" Make sure you have enough. You need about 3 times more doors than villagers for them to even consider breeding. This was the sole factor in breeding in previous versions. Now, it's only the first part.

Also, my intuition says the chunk that contains the new village needs to be loaded for breeding to occur. This means you need to be close to the villagers for longer periods of time.

Lastly, they need to be close to each other to breed.

Hope that helps!