How exactly is "true randomness" defined in the realms of cryptography?
Randomness is not a property of strings of bits (or characters of any sort).  Rather it is a property of the process that generates those strings.  However, it is convenient to conflate the string with the thing that produced the string, and thus to speak about strings being “random” or “not random”.

The string 00000, for example, is random if it was the outcome of “random process”, such as a coin being tossed five times and landing on tails five times in a row.  Similarly, the string 1,2,3,4,5,6 is random if it was the outcome from rolling a die six times.  Note, the random process does not need to be “fair” to be random, though processes that deviate substantially from the uniform distribution are not as useful for cryptographic purposes.

What is a “random process”?  As I think about it, a random process is either an indeterministic process (if any actually exist), or a deterministic process for which the entropy of what we don't know (that is germane to the outcome) is greater than the entropy of the generated string.  There is a lot we don't know (and can never know, given the Uncertainty Principle) about the state of the flipped coin – e.g. the exact position and momentum of every particle in the coin, the air around it, and the hand that flips it (all of which are germane, in that we presumably would need to know them to accurately predict the outcome of the flip). 

If the string 00000 was the outcome of a number-generating algorithm, where the entropy of what was unknown and germane about the algorithm was less than 5 bits (e.g. we knew the algorithm and all but four bits of information about the seed), then the string would be “not-random”. At best it could be “pseudorandom”, meaning computationally difficult to distinguish from random.