I don't see any studies of this sort on prime numbers, though I'm sure you could conduct an informal one and get a good estimate relatively quickly. Instead, I tackle your final note:
A good answer would be numerical data about this question or a similar one.
How about: Is zero even?
Citing a popular media piece:
According to Dr James Grime of the Millennium Maths Project at Cambridge University, reaction time experiments in the 1990s revealed people are 10% slower at deciding whether zero is odd or even than other numbers.
Children find it particularly difficult to recognise if zero is odd or even. "A survey of primary school children in the 1990s showed that about 50% thought zero is even, about 20% thought it was odd and the remaining 30% thought it was neither, both, or that they don't know," explains Dr Grime.
And it gets worse:
It's not just the public who have struggled to recognise zero as an even number. During the smog in 1977 in Paris, car use was restricted so that people with licence plates ending in odd or even numbers drove on alternate days.
"The police did not know whether to stop the zero-numbered licence plates and so they just let them pass because they didn't know whether it was odd or even," says Dr Grime.
As for primes: Are you wondering about the percentage of people who can precisely define when a number is prime? If so, I think asking them if $0$ or $1$ is prime is probably enough to reduce this number down to something miniscule.
(Consider this popular media piece on the polymath project post-Zhang, in which the author writes: Over the past year, mathematicians have been battling it out in a game involving primes, numbers that are only divisible by themselves and 1. Even allowing for "numbers" to refer to natural numbers, this definition would still include 1 as a prime.)
In any event, I think it is nice when popular media pieces on mathematics start with some basic definitions. I don't even think it's a stretch to include a definition of prime numbers in a graduate textbook on number theory.
In the most recent issue of the AMM, there is an article entitled The primes that Euclid forgot (link). Though 'prime' is not explicitly defined, 'quadratic residue' is: see the start of section two. Of course, this latter term is sure to be quite unfamiliar to the general populace; but I expect it is known by a very large proportion of the American Mathematical Monthly's readership. If the authors there feel including a reminder-definition is worthwhile, then I suggest you dedicate a line to defining the term 'prime' in any relevant popular media piece.
