So the Guardian said Christmas and Birthday cards, but the Judge said the order would say he could see you if he wants. He is 13 and appealing could take a long time with ISW reports etc.
So don't see it as walking away. Instead, just always be available in case he decides to come. Maybe just accept that it is up to him now. Things will change. Now the court case is over, your ex won't bother to keep trying to alienate him - she's won her battle so there's no more fight. He may start to get his head clearer and think for himself. And he may wonder about you and miss you.
In addition to that, 13 is a really funny age - they are hormonal. As I mentioned earlier, by 15 he will be very different. What has happened in some cases is, once it's all over, the ex actually offers you to have him sometimes - she might not be able to handle him. Or she might fall out with him.
So as mentioned above - keep an online presence. Don't bemoan in it though - post things about the good, interesting things you're doing in life. Start a new project, join a new club and post about new friends, hobbies, interests. Sound upbeat and cheerful and even jokey.
Because your son will probably look you up sometimes - and he'll see someone very different to the picture your ex has painted, and may decide to get in contact with you.
Uncertainty is not nice. I thought I'd lost my son when he was 14 - not only was the ex keeping him away and taking him away a lot but he had aligned with his family there. Once he realised he was hardly seeing me though he changed. Especially by 15. He ended up moving here eventually.
He has probably been under huge pressure from the ex throughout this process - threats, promises and all sorts. To agree with her. Threaten to take a 13 year old's phone away and they'll do anything! But he will start to think more and may feel resentful at what she has done - to him, if not to you.
He will be getting more interested in friends than parents over the next couple of years.
So rather than seeing that you've lost your son, just think of it that he is away for a while. I know that might not help much, but teenagers can cause a lot of headaches - focus on building something new and new achievements in your own life and he might come to you of his own accord.
He has been caught in the middle, and now he isn't. So it'll be a relief for him. He maybe genuinely did feel he didn't want to see you, just to take the pressure off from being caught in the middle. Once that pressure is off things could be different.
Don't expect anything, but don't give up hope either. You know he's out there and he knows you're out there. He's not a little boy any more.
Does he have any other family at your ex's? Half siblings etc? He may well get sick of her. She will trip up probably at some point and he'll see some things were lies.
But he will want to spend most of his time with friends as well.
It's hard when there is uncertainty as to what will happen - so you just live every day and keep positive that things will work out well in the end - however that happens.
You will have shared memories - he will have those too - they will come to him now and then. Don't give up hope. But do get on with life and start new things.