Deposit request from Glasgow

They are making you take the risk rather than them, or at least risk sharing. So you have to weigh up whether you are prepared to risk £300 or your place at Glasgow. I doubt they will give you a deadline by which to pay the deposit, because the terms of the deal are first come first served. That is usually incompatible with a deadline.

What course is it? Unless it's very popular, you should be alright. I can't imagine most courses at at Glasgow would fill up this early and you won't be the only one reluctant to pay it so soon...

I don't understand how this works.. do they give out more offers than the places that they have and then take the people who pay first? It's unbelievable!!

All universities over-offer for their postgraduate courses, because there are a lot of people who can't take up their places unless they receive funding as well as a lot of people who have received more than one offer and choose to go elsewhere. The only thing that's a bit unusual is the 'first come first served' bit, which seems to be a slightly underhand way of forcing offer-holders into making a quick decision.
Even if it's a popular course, though, I don't suppose it would really be in the department's interest to over-offer massively beyond what's needed to make up for the places that don't get taken up through lack of funding or people choosing to go elsewhere. It would only mean unnecessary admin work for them, and if they did turn away vast numbers of offer-holders for not paying the deposit right away they'd be likely to end up getting bad press for it (which presumably they'll want to avoid). So I'd say there's still a good chance that taking a couple of weeks to make up your mind won't cost you your place - after all, there'll be a lot of other people in your situation and they won't necessarily want to rush their decision any more than you do. It's impossible to be certain, though, of course...