At the top level what you say is more or less correct, John but the problems occur when you dig below that level.
I say "more or less" because I don't think "deems fit" is quite the correct description. I think it is pretty much in proportion to population size possibly modified by demographics. For example, if the decision is to give it to old people and one country has a bigger proportion of old people they might get slightly more doses than their relative population size would suggest.
I think which groups the vaccine is given to depends on what the results of the clinical trials reveal. For example, with the vaccines that have reported so far, they appear to prevent people developing advanced symptoms, they appear to work well with the older population who are most vulnerable but it is not certain how well they prevent the disease being passed on. If those conclusions are upheld then it is best to give them to the most vulnerable groups and that is the preliminary recommendation in the UK. If, however, it turns out that they are very good at preventing the spread of the disease then it might be better to switch the priority to the working age population so they can go out to work unhindered. It is also possible that one vaccine might be better for the older group and another vaccine better for the working age population. Different vaccines work on different principles so it is quite possible that they might work better with different groups. I think the ones that have presented preliminary results so far all work using mRNA and simulate the virus spike protein so they are likely to work in very similar ways.
Apart from specifying which groups should be given preference I don't think the EU will get into details, that will be down to the individual national governments to decide based on how their particular healthcare systems work. I cannot imagine the EU deciding that Fred Smith at 23 Railway Cuttings Heraklion should get it and his neighbour Bert Jones shouldn't. That is where uncertainty lies. How will the Greek government decide and what process will they use?