What makes a shirt a sports shirt, which is to say something other than a dress shirt? Two things: fabric and pockets.
A sports shirt is usually intended to be worn as a top garment without a jacket, or, less often, under a sweater. That means it needs to take on some of the functions of a jacket. And, in my book, that means a sports shirt starts with two pleated pockets that button.
A sports shirt can have short sleeves or long sleeves with barrel cuffs. It may have shirt tails and a buttondown collar. Or, it may have square tails and a camp collar, which is a collar without the reinforcing band that keeps the collar standing. But when he's wearing a sports shirt a man needs pockets (it's the opposite of a dress shirt which in my opinion shouldn't have them).
The other characteristic of a sports shirt is that it typically has bolder fabric than a dress shirt. That might be bright solids in linen or silk; or Tattersalls, Scottish tartans, cotton madras or other checks and plaids. Heavier weights can also come into play, such as denim, viyella (like the pictured shirts from Ben Silver), moleskin and twill.
Whatever the fabric, remember that one pocket is not enough. There must be one on each side for balance across the chest. And buttons to keep them closed, so your sunglasses don't fall out every time you lean over.