That is a good way to think of it. Indeed, that is how cryptographic randomness is typically defined.

Simply put: given the first k bits of a random stream, can you predict the k+1th bit (more than 50% of the time)?

A generator that passes this test will pass any statistical randomness test, but the converse is not necessarily true. For example Mersenne Twister is a good generator from a statistical perspective, but it's actually quite easy to recover its internal state by observing a small amount of output. (Around 20k bits, if I recall correctly.)