If you're talking about big corporate graduate schemes I would've thought classification is important to get a foot in the door. The big employers have a lot of CVs to sift through and need differentiators. I can't say for certain because these things are multidimensional and I don't know of any empirical studies on this.
You don't have to work for one of those glamorous big ticket companies, it depends where you want to go in the future. You can also get great experience working for smaller companies; they tend to develop you in more areas as you can get more exposure to different parts of the given organisation so long as you are willing (actually, flexibility is often a requirement!). You can also get good entrepreneurial skills this way.
So four or five years down the line the classification gets far less important and people begin to look at your achievements since graduating.
You're doing the right thing in thinking about the possibilities. I started looking around late in the first term of my final year too and I ended with plenty of job offers. Sounds like quite a few of you are on the ball, clearly students have pulled their socks up in the past decade or so