Is Minecraft considered to be an MMO?

On some forums that I frequent, someone has been taking about Minecraft and they posted it in the MMO sub-forum. However, I don't view Minecraft as being an MMO. I'm going to post the discussion I had with the person.

Me:

"Minecraft isn't an MMO, it's a game that has a Multiplayer aspect. An MMO is a game that's connected to the internet all the time and to keep playing it you'd need an internet connection consistently. MMO's don't have a choice between single player or multiplayer.

If Minecraft is an MMO then other games that have Multiplayer aspects are as well. Like GTA Online except they aren't. Because MMO's don't have a set amount of how many players can come on at one time. Minecraft and GTA Online both have a minuscule cap on how many people can play at any given time."

Them:

As far as I'm concerned those games which have multiplayer online are MMO's. The cap is superficial, it only limits how many ppl are on a server, even large scale MMO's have caps, look at World of Warcraft, they limit the number of ppl per server. The servers are online all the time/i], just like with any other MMO, they don't stop being there when you are no longer connected. Just because a game can do more than one thing doesn't mean it is any less either of those things. That's like saying a game can only be an RPG or a shooter it can't be both.

I don't believe that games like Minecraft, GTA (5) Online, Diablo 3 are MMO's. Because MMO's don't have a choice in what type of mode you want to play. (I.E. Single-Player and Multiplayer) MMO's only have one type of mode. Who do you think is right?

Added (1). I forgot to add quotes to the word on the opposition. Starts below the word them.
"I don't believe" is also apart of my viewpoint and question.

The reason it's not an MMO is because it is not massively multiplayer. MMORPGs do have server caps but they are usually somewhere in the hundreds or thousands of players. You don't have 1500 players on a Minecraft server. Therefore it is not massively multiplayer. That's the entire distinction.