[RPG] Which attack yields higher average damage

The answer to your question is No, using versatile weapons with two hands does not disqualify them as monk weapons.

But not because of any reason that made you ask the question, because your whole question is wrong as it's clear from your question you haven't understood the paragraph in question.

The paragraph in question on pg. 78, PHB says (emphasis mine):

At 1st level, your practice of martial arts gives you mastery of combat styles that use unarmed strikes and monk weapons, which are shortswords and any simple melee weapons that don't have the two-handed or heavy property.

Let's look at what that paragraph really says.

The first thing it does is stating that at 1st level a monk has a master of different combat styles using unarmed strikes and monk weapons.

The second thing it does is stating that monk weapons are shortswords and any simple melee weapon that doesn't have either the heavy property or the two-handed property.

That's everything that paragraph states. It simply tells you that monks fight unarmed or with monk weapons and then defines monk weapons as either shortswords or any simple melee weapon that isn't heavy or requires two-hands to use.

Simple melee weapon with heavy or two-handed property = not monk weapons.

Simple melee weapon without heavy or two-handed property = monk weapons.

So what about the versatile property? A weapon with that property just gives you the option to use two hands during an attack for greater damage. However, in this case whether a weapon has the versatile property or not has no impact on the RAW as it only purpose in this case is to define what a monk weapon is, which is a shortsword or any simple melee weapon without either the heavy property or the two-handed property. Any other properties such as light, finesse, reach or versatile has in this case no bearing on the rules. Nor does how you use said weapon.

The monk's Martial Arts feature reads:

When you use the Attack action with an unarmed strike or monk weapon on your turn, you can make one unarmed strike as a bonus action.

On page 195 of the PHB, under "Melee Attacks", it says (as of the first PHB errata in 2015):

Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head butt, or similar forceful blow (none of which count as weapons.

That, to me, makes it sound like you could attack with a monk weapon using two hands, then make a kick or head butt (or another forceful blow with a part of the body that isn't your hand) as your bonus action.