I've been researching the senses for some time, as an amateur. The non-skin external senses are pretty straightforward: sight, hearing, smell, taste, and what scientists call vestibular sensation. (This last includes two pieces: one senses the direction of gravity, the other senses angular acceleration. Since gravity is itself a form of acceleration, I think of the whole sense as the sense of acceleration. The vestibule is the part of the inner ear that doesn't do hearing.)
So this is five mostly straightforward senses. You can increase the number if you count the visual contribution to the body's clock separately from sight, or pheromone sensing (if any) separately from smell, or if you split vestibular sensation into two, but seems to me those are cheating.
Balance is something the brain does with (mostly) sight and vestibular sensation. Proprioception is something the brain does with (mostly) the muscle senses and vestibular sensation, though it also lets the skin senses and sight pitch in if they can. I wouldn't consider either a sense in its own right.
Thermoception is, among other things, a skin sense. And this creates trouble because there's no widely accepted definition of what separates senses, so it isn't obvious how to separate the skin senses either from each other, or from the muscle and visceral senses.
For starters: There are four kinds of receptors we ordinarily understand as "touch" even by narrow definitions. Two of these are also found in the muscles, ligaments, and joints. Meanwhile, we have two very different kinds of receptor for heat and cold. One responds to heat; the other responds mostly to cold, but also to some levels of heat. (This is why cold can sometimes feel like it's burning you.) These may or may not be processed separately in the brain; and I have no idea whether, or to what extent, they feed into the somatosensory cortex, the touch map of the body, which is what I came to this page hoping to find out. So is thermoception one sense, two (separate receptors, perhaps separate brain processing), or zero (part of touch because it hits the somatosensory cortex -- if it does) ?
Don't even get me started about pain. For example, the top level of hearing according to hearing specialists is the "threshold of pain". Is this kind of pain the same as the kind we sense with our skin? With our guts? How about the pain from overly bright light?
And the fun only continues. The article blithely asserts that external thermoception works very differently from internal thermoregulation. Well, duh. But does external thermoception work differently from stomach thermoception? Yes, our stomachs sense heat and cold.
Once you get inside the body, also, things get very confused with the chemical senses. Many of our glands can sense when a particular chemical is low or high in a particular fluid. Is each of these a separate sense?
I have no idea how you get to 21 senses, but if those are all external senses, then whatever classification it is, ought to include, for example, each glandular sensor as a sense, and probably ends up somewhere in the hundreds. --Joe B.