should i make the first move?

Of course you should talk to her. How else can you find out if she is willing to put aside your past and start anew.

Start out as friends and see if you can build on that, if that's what you need. Otherwise, move on.

On the one hand:

... this is quite a positive sign that she's willing to or at least partially forgiven you for whatever happened to cause the falling out and she still wants to be casual friends

But on the other hand:

This is a recipe for disaster and it could ultimately kill all chances you have with ever making things right and becoming her friend or more. Since I am certain you DON'T have ESP or the ability to read minds, you DON'T have the right to assume such things as you already wrote ...

From what you wrote here? It seems she only wants to return to a casual friendship for now. Don't burn your bridges by trying to convince her that you believe she "doesn't like him" even if she doesn't have particularly strong emotions towards him in the first place. It's her call on whether she is in a relationship with this new guy or not despite the level of involvement. If you persist down this road as if you will try to convince her that her feelings for this new boyfriend then it'll just show that your attempt at being friends is disingenuous as you want to possibly skip from friend zone status into full boyfriend with benefits mode which let me make clear is wrong as its quite manipulative thing to do.