Warning: I am not a native speaker of British English. This may be wrong.
I am under the impression that:
a pram is a type of baby-conveying cart in which the baby is laid down, such that it is facing up and backward from the direction it is pushed; in this way the person pushing the pram can see into it and look the baby in it in the face.
a pushchair is a type of baby-conveying cart in which the baby is seated upright, such that the baby faces forward (in respect to the direction it is pushed). The baby/passenger in the pushchair faces the same direction as the person pushing them. Such pushchairs often have a lot of baby-containing structure, padding, and onboard features (food tray for the infant, cup holders for the pusher, e.g.) and can even be made almost entirely of rigid material. (Popular in the US right now is a type of baby-conveyance in which a crash-worthy hard-shell baby seat comes with a snap-into mounting for use as a "car seat" in automobiles and a snap-into frame with wheels for pushing as a pushchair, such that one can hotswap the whole infant-seat assembly between your car and your pushchair, without having to unmount the child from the seat.)
a buggy, by the definition you found, would sound to be what Americans call a folding stroller: like the pushchair it faces the passenger outward and has them seated, but is built like a typical lawn chair, of metal tubing and fabric stretched between to form the seat and back.
P.S. Just in case that wasn't confusing enough: apparently there is also some sort of hybrid pram-pushchair thing, that allows one to configure it in either the baby-faces-up or baby-faces-forward configuration.