How to the Saudis check who is a muslim when entering the cities Mecca or Medina and what is there definition of a muslim.

Probably ask some questions, the most basic being "are you a muslim", possibly followed up by a request to quote a part of the Koran.

I think non-muslims are allowed in Medina.
I assume tourists apply for a visa?

Non muslims are not allowed in Medina, and I don't think the Saudi government issues tourist visas at all.

You provide proof that you are a Muslim when you apply for a visa

What kind of proof though? A letter from your imam?

A quick google suggests Saudi Arabia do issue tourists visas and non-Muslims can enter medina (but not the very central part)

According to a website about the process:

"If someone has a non-Muslim name, he or she should submit a certificate from a mosque or an Islamic center stating that the applicant is a Muslim."

A quick google will tell you SA issues visas for business purposes but you must have have an official invitation to travel there for that purpose. It doesn't issue tourist visas per se but does issue visas to those travelling to visit relatives or friends.

As for Mecca and Medina there are apparently sign posts which state non-Muslins are not permitted to enter.

Oh you're right, the Saudis did start issuing limited tourist visas in 2013, then cancelled them in 2014 and said this year they plan to start issuing them again. Suffice to say they're not exactly enthusiastic about throwing open their doors to outsiders.

Seems I had the right idea!

Pretty much.

I asked someone about this earlier today he said that when you apply for a visa or residents permit you are asked what religion you are. If the have doubts about whether you are a muslim you will be asked to provide a reference or conversation certificate from a imam/sheikh in your mosque.

when you arrive at mecca or madina checkpoints the cops ask to see your visa or residents permit.although I'm not sure what they do if you are a Saudi Arabian citizen. If a non muslim if found in the holy cities he will be deported and banned form the country.

Although he did say once a italian ambassador was seen in mecca when the cops arrested him he said he was muslim but when there Were different versions of his story when questioned at the station.

What about other types of cotton?

Virgin cotton of course. You didn't think the martyr's reward of seventy two virgins were human women did you?

Nope, virgin wool sock puppets (so the martyr can slip his hand inside).