Search:

Try a razor on it, or do a quick lapping of the other side to see of the spots go through. It is not uncommon for a natural stone to be dramatically different from one side to the other. One inch...

As long as it’s smooth.

I think the better find may be the Yellow/Green Thüringen which I think you may have there.

One of my 2 dollar Swap meet finds is a Y/G Thüringen and one of my best...

How deep is the hole?

I would do a mm at a time, a mm of dust and couple drops of CA. Let cure then more Dust, more CA. if you have enough dust to fill.
Make sure the stone and dust are dry.
...

The photos with the glue are the ends? I think I see saw mark under the glue.

The Green photo does look like a Thüringen with slurry, what’s the other side look like?

The hole in the BBW may not matter as long as it does not snag. It looks as though it was use with the hole. The hole looks filled with slurry.

You may want to clean the stone and work on a cookie...

Yes the first is a Coticule with a BBW back. Lap the back (BBW) flat you may still be able to use it depending on what they glued it with.
Yes, Thüringen slurry can look white on the stone and grey...