I thought all airlines stopped issuing paper tickets a few years back. The electronic systems were built around the old terminology though, so talk of coupons and tickets etc doesn't always make sense unless you can remember the good old days of red carbon!
It does mystify me why some airports keep the small bit - keeping the big bit makes sense as that was what would hold the coupon. I'm not sure why BA doesn't keep a portion at the gate - even just as a further check in numbers should the count not match as I'm sure scanning barcodes isn't 100% reliable.