I think that the definitions are somewhat elastic and depend in part on who is defining them and for what purpose. For example the largest motor insurance company in my neck of the woods (NRMA) offers insurance for the various types of historic cars as defined in their booklet:
We offer insurance for:
veteran vehicles made before 1919
vintage vehicles made between 1919 and 1930, and
classic vehicles that are 15 years of age or more (and recognised by us as being a collectable vehicle)
It's worth noting that the NRMA's definition of "Veteran" is in accordance with the one that Stu W cited from the Collins dictionary. (Which also agrees with the OED's one.)
Also, NRMA's definition of "vintage" is in accordance with the OED's definition.
Although I can't find a reference to this at the moment my Mk I memory seems to recall that a similar distinction between vintage and veteran once existed with the motor registering authority as well, though it was based on how many years ago the cars had been manufactured relative to the registration date. However this was many, many years ago. This week's authority (the Roads and Maritime Service, who knows what it will be next week) just has an umbrella term "Historic Vehicles":
Historic vehicles may be registered under the Conditional Registration
Scheme. The conditional registration of historic vehicles is referred
to as the Historic Vehicle Scheme (HVS). The HVS was developed in
consultation with industry.
Historic vehicles (including trailers) must be 30 years of age or
older as from the year of manufacture.
Interestingly this would make a Chrysler Sigma historic which... er, yes, quite.
In any case my point is that you may well find that different insurance companies and motor traffic authorities use different definitions, but the NRMA / OED / CED ones seem to be be useful as a general guide.