I think "random" does mean something more than "unpredictable". It means something more like "independent of things you care about". More precisely, that's what it should mean in most places where it's used.

(I'm not quite satisfied with this formulation; e.g., a "random" thing that always takes the same value is independent of everything, but you wouldn't usually want to call it random. What we're really trying to get at is "statistically indistinguishable from idealized randomness" but it would be nice to find a way of saying it that doesn't appeal to an existing notion of randomness. Perhaps something like "incompressible, on average, given the accessible state of all the other things we care about".)

Imagine, if you will, a lottery that works as follows. Each lottery ticket bears a SHA-256 hash of (the ticket's lottery numbers + a further string); the further string is not revealed until the time of the draw. When drawing time comes, the winning numbers and the further string are revealed on national TV, and if you think you might have a winning ticket you can bring it to have the hash checked.

In Scenario 1, the winning numbers are chosen "at random". In Scenario 2, you (and only you) have the magical power to make the winning numbers be the ones hashed onto your ticket. You don't know the further string. You also don't know what your ticket's numbers are. You just know that after you perform your magical ritual the numbers drawn will match the numbers on your ticket.

The numbers drawn are still unpredictable. No one knows what numbers are on your ticket. (Let's suppose that the tickets are made by some process that after making each ticket erases all evidence of what numbers have been hashed onto it.) But something is predictable, namely that the numbers drawn will match yours and you will win the lottery.

The lottery numbers in Scenario 2 are still "random" in some sense, but this is exactly the kind of situation you're trying to avoid when you deliberately randomize things.

So, the answer to Romashka's question is: those results would be less random-in-my-sense because they might correlate with interesting properties of the people in the experiment, which could be a problem if e.g. you were using these coin "tosses" to control something you're doing with the people (e.g., which ones to use for the next phase of the experiment, or which of two things to ask them to do). They might well also diverge somewhat from the 1:1 ratio you'd expect from theoretical unbiased coin tosses (e.g., maybe most people prefer to put coins with their "heads" face upwards).