It will take a great deal of pressure to get the section of Planning Dept which deals with this to take cycling seriously in Princes St or George St. For example, see some of the history re the Gehl report.
Note too that the Scotsman article doesn't refer to Princes St at all, and it only talks of 'parts of' George Street. I suspect the relevant section of the planning dept is thinking of widened pavements/cafes in some stretches, with nothing for cycling except perhaps a bit more bike parking. That is certainly what they were talking about a year or two ago, and the Scotsman article doesn't suggest any change to that.
There are councillors and officials who'd seriously like to see much more done to improve cycling conditions in the city centre, even at the expense of cars, but in the past they have been in the minority or unwilling to speak up strongly. I'm not convinced that has changed until we see some actual plans, timescales and funding which prove it!
Victoria and Cockburn St are hardly major cycling routes, though obviously any car restrictions there would be welcome.
Chambers St - this is nothing new, it's in the Council's Southern Arc plan which there was a massive consultation on a year or two ago, but if the chambers St project is in fact going to come on stream soon that must be good - again the devil will be in the detail.
So overall, nothing new that's really good for cycling in the central area (particularly Princes St/ George St) is likely to happen without a great deal of continuing pressure to get more councillors to be really serious about cycling provision.