As I understand from other things I've read on this topic, it should be that if the lines touch or overlap, then the players will be judged as level and so there would be no offside.
I think you're right that up till now, they'd been trying to 'force' a decision by using the thinnest lines possible and even moving those slightly to ensure a distinction between them. For instance you could sometimes see as they showed you the VAR footage, that they would start with the lines in one position but then adjust both the lines on the ground and the lines coming down from the players' bodies, to ensure there was a gap between them. I don't recall them ever drawing two lines that were absolutely on top of each other, even though we knew that occasionally, that should have been the case.
They must have a specific way of working it out already in mind though, because Mike Riley when he was talking about this, was able to say precisely how many goals that were ruled out for a marginal offside in the Premier League so far this season, would have been allowed to stand if thicker lines were in use.