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AlephNeil comments on Conditioning on Observers - Less Wrong
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To an outside observer in the first scenario, the probability of a particular awakening being morning, picked at random, is 6/7.
To an outside observer in the second scenario, the probability of a particular awakening being morning is 1 in the tails case and 0 in the heads case, and each case is equally likely, so the probability of a particular awakening being morning is 1/2.
So adding the coin flip does change things about the scenario if you're an outside observer, so I would not be surprised to find it changes things for the subject as well.
In my use of the words 'every week' I am implicitly--I take that back, I am explicitly supposing that every week the procedure is repeated.
So we would obtain an indefinitely long sequence of awakenings, of which 1/7 are in the evening and 6/7 in the morning.
For any such finite sequence of awakenings, there would (when viewed from the outside) be a 50% chance for a particular week of waking up in an evening, and a 50% chance for a particular week of waking up in the mornings - you can then assign a uniform distribution for particular weeks, getting a 1/6 probability of a particular morning in a tails week. If you pick an awakening randomly on that distribution, you have a 1/2 probability it's an evening and a 1/12 probability it's any particular morning (ETA: out of the week).