Hello and welcome
Well done on the healthy life changes you've made so far
Is your goal of 18-14 stones possible by Christmas? Well, I don't think it's quite impossible but I think it is unlikely you will lose 4 stone by then. To give you a rough idea I was 19 and a half stones back in March when I restarted trying to lose weight and am now 16 and a half stones. Getting this far has taken 50 minutes of an exercise dvd up to 6 days a week (was less at first) and cutting calories to an average of 1400-1500 (probably a little low for my weight especially as some days I end up with less though some days I end up with more) and drinking between 6 and 10 glasses of water a day. I've had to be very dedicated about these things and will have to for a very long time to keep up with anything like the rate of weight loss I have had.
I would probably aim for about 16 stones by Christmas (and any more in a bonus). I'm currently aiming for 15 stones by Christmas (though may make that lower - maybe 14stones - if my weight loss doesn't slow down). 1-2lb a week is considered healthy weight loss which means roughly 20-40lb is very realistic if you work hard all the time between now and Christmas. You will probably lose more the first few weeks though.
As for how to lose weight, I calorie count. It means I can eat what I want as long as it's within planned calories though I try and keep refined carbs low most days and try to make healthy eating choices. Have a look on the calorie counting forum more more information. With a lot to lose you are better to start exercise too if you aren't already even if it's just increasing how much you walk.
You say you eat the same as your thin friends. Are you sure you really do? Check your portion sizes - maybe you eat more breakfast than they do, or more at a different meal than they do. It's so easy to give yourself slightly too much and that can mean the difference between maintaining weight and losing weight. If you really are eating the same as them it may also be that you aren't eating enough. Not eating enough could be making you want to binge. If they are 9 or 10 stones and you are 18 stones your body needs more calories at this stage than theirs. If you write down everything you eat over the next week you'll be able to see where you can make healthier choices, where you're eating too much and where you are eating too little.
As for the binge eating, there's a forum here for that where you can get extra support. I am also a (recovered I hope but certainly recovering) binge eater. Yes stopping binge eating makes a big difference to your plan being successful. You should find a lot of helpful information there.
Read all the information you can here - everyone's very nice and very supportive and you will learn a lot form the weight boards. Remember any weight off is better than it being on so even if you only lose 1stone by Christmas instead of 4 that's going to make a big difference to how you look and feel. You'll find most people here work in lbs rather than stones except for those of us in the UK
If you too are in the UK then come and join us on that forum also