That definition does have the merit of simplicity.
But then "classical" in classical transport does not have to be
specified (since it is just ordinary). And we think all possible
forms of transport and communication are limited by the speed of
light, which further simplifies the definition: classical transport
is just transport, and classical transport of information is just
transport of information!
Still I don't think it can be correct. More likely seems this:
1) Classical transport (of information) obeys the Bell inequality
for the amount of mutual information between receiver and sender
after the transport. With signaling limited by the speed of light.
2) Ordinary transport means any QM-allowed transport, so the limit
is now the Tsirelson bound (which for a qubit is sqrt(2) times
higher than the Bell limit. Signaling is still light-speed limited.
3) Supra-quantum transport is for example the Popescu-Rohrlich 
behavior, with for the correlations again a factor sqrt(2) higher
bound (so already a factor 2 higher than what "classical" allows!)
But even that example only allows light-speed-limited signaling.
4) More extreme supra-quantum transport, i.e. even further away from
currently accepted physics, might even (finally) allow superluminal
signaling. But with that we would be digressing far from the topic..
It seems to come down to what "ordinary" means. If that is case 2),
than logically case 1) would need specifying by a qualifier like
 See "Popescu-Rohrlich box"