Framing basement ceiling with concrete slab

Masonry-screw furring strips (1x3 or similar, flat) to the ceiling, screw foam to that (barely) or glue foam to that, and screw drywall through the foam to that as well (throughly). You have all the structure you could possibly need in the form of the slab - all you need is something to hold screws, unless you can find drywall screws that go into concrete, which I'm not aware of or I'd suggest using those. 2x6 framing seems utterly pointless and a waste of space.

With the right glues you might be able to do the whole thing with glue (foam to slab, drywall to foam) but I'm not confident that it's currently a reliable method, and it might be slow holding sheets up while waiting for the glue to cure, anyway.

Install a suspended ceiling.

attach eye anchors to the masonry and use wires to suport the channel which supports the drywall

pack your batts above the channels then attach the rigid foam and the drywall using screws.

You could run 2x6's under the ceiling side to side by installing a ledger on either side and using joist hangers. This will be far less work than trying to attach to the ceiling, and you don't have to worry about compromising the ceiling, fasteners rated for overhead use, etc. You can use the same types of fasteners used to attach ledgers for decks to concrete walls. That will be more than adequate.

Just make sure you have a plan for moisture to escape, and use pressure treated lumber where ever it's in contact with the concrete, this might be the trickiest part of the job. Strapping under the ceiling joists might be a big help.

One thing to watch for. Pressure treated lumber is sold very wet and green and shrinks quite a bit. It would be tempting to use non-treated lumber for the ceiling joists, but I see wood in that space getting pretty damp even without direct contact with the concrete. This might be another advantage of the strapping, it will allow for some movement of the wood without messing up the drywall.

It all might be enough to make you want to install a drop ceiling - nobody installs them because they look nice, but they do make a lot of things simpler.