> What does "strategy" mean here? I might just happen to have a strategy which involves betting on the exact sequence of heads and tails in a given sequence.
That's a very narrow program.
> The analogy in terms of languages is that my language might just happen to have a short keyword that represents a given sequence of heads and tails.
The sequence still needs to be generated "somehow". Either by executing the program and producing the sequence, or by explicitly stating it. Even if you have it "cached" and "represented" in your language, you still need to generate the sequence. The resources spent here is the Kolmogorov complexity.
The easiest way to expand your little program is to say that you have a seed s.t. any consecutive generation results in a consecutive sequence that matches up to the period of the generator. Now it is more generic, but has a period. You can then expand this to accept multiple seeds and once it has reached a period, to simply take the next seed.
Should this sequence be finite, you are in luck. Your program can have length O(generator + N/P) where N is length of sequence, and P is the period of your RNG.
All this is is just compression which plays into the whole Kolmogorov complexity.