Hi there! If you are considering joining a reenacting unit, I cannot recommend strongly enough that you check with them regarding standards. What different organizations consider accurate can vary wildly, with some taking the "close enough" stance and others insisting on hand stitching and vegetable-dyed fabric and all that. I really don't want to be responsible for you buying something that turns out to be unusable, so if it's alright with you, I'm going to avoid commenting on any particular vendor's products and confine myself to the history.

The issue with gambesons is a lack of evidence for their existence in the early period rather than any evidence against their existence. There is a tantalizing Roman reference to a garment called a subarmalis, and we have quite clear artistic depictions of gambesons and aketons from the 13th century on, but not much in between. I think they probably wore them, but that can't really be substantiated. If they did, they would be of linen, wool, or leather, not cotton, as in the link you included.

Your stereotypical knight of the 12th century would have worn a knee-length mail coat, made from a mix of riveted and solid iron rings. Early in the century, three-quarter or full sleeves would have been common, while by the end of the century integrated mittens would be more the norm. A mail coiffe was probably integral to the hauberk as well. He probably, but not necessarily, wore mail leggings strapped on above the knees. The helmet would almost certainly be of the conical or round variety, with a pronounced nasal.