How does this not trivially exclude everything (the first definition, not the circular restatement)? Considering that, say, a sequence of coin flips either has the property of "beginning with heads" or "beginning with tails," it certainly appears to.

My bad, I was being too vague - sorry for that.

I meant "any atypical property" instead of "any property".

An "atypical property" being a property that almost no sequence has, i.e the set of sequences which exhibit this propery is a measure-zero set.

In a practical sense, I'd say the definition is simply our inability to recognize a pattern at this point; it's completely subjective.

Lehmer (1951), by way of Knuth Vol. 2, "A random sequence is a vague notion embodying the idea of a sequence in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests, traditional with statisticians and depending somewhat on the uses to which the sequence is to be put."