I actually came into it not knowing any more than you – and I tried all those you mentioned. I painted the white, then tried to use painter’s tape – no good, the bit hit the tape, and left ragged edges that then the vacuum pulled up (and I really can’t recommend trying to cut MDF and especially not Extira without a vacuum going). I then tried contact paper – it had more adhesion to the wood, but still, no good – if the bit hit it just right it ripped whole sections up. I then tried spray adhesive along with contact paper – it worked, but left a mess on top, so that was out. If you were JUST carving letters into wood, and then spray painting – contact paper works. Try to do a pocket, and text standing up in that pocket and the paper just doesn’t have the adhesive to hold onto the small surface of the letters while everything around it is getting cut down.
Then, I came across this video that told me exactly what I needed to do – I modified the technique slightly to use a ultra smooth FOAM roller (you definitely want foam), and I’d get a little paint on it, and roll it back and forth until it almost seemed like it was barely painting. Then, I went over the top (you’ll do several coats, with sanding in between, but thin – you want no running at all). So, I did the pocket color first, and THEN the top white part. It sounds like it wouldn’t work – but watch the video and see how he does it with a foam brush and how well it works for him.