Cut a 2″ thick marble slab theself

A diamond hole saw will do the job, use plenty of water and light pressure.

You will do better with a single hole, AFA breakage goes, compared to making 2 edge semi circles.

Update

If we are talking about a toilet flange, there are 2 methods, each with variations, depending on your equipment.

1. With a tile wet saw, raise and lower the blade such that you cut diameter lines edge to edge. Rotate the tile 10 degrees and repeat. Flip the tile over and repeat from below. Remove teeth with an angle grinder fitted with a diamond blade. If you are connected to a GFCI outlet, a helper can slowly spritz the cutting with a water bottle. It will keep the dust to a minimum. I would recommend a continuous edge (not segmented or stepped) diamond blade for this. The cheap table type tile wet saws actually excel for this cut, allowing fast and accurate cuts, all from below.
2. The whole cut can be made with an angle grinder setup described above, you just have to tip the blade at 45 to the surface of the tile and sneak up on the line from the inside of the desired cut out. With marble you have to go slow and minimize pressure. This method also works nicely for electrical boxes.

Ahhhhh no wet saw for small glass mosaics. First it will bounce all over when cutting and second you might get a nice chunk of glass in you.

You will be cutting everything one mosaic tile at a time. Take it off of the mesh.

#1 Make sure when laying out your tile that your cuts are in the corners or in less seen areas.

#2 If you need to cut the corner of a tile or odd shapes use glass nippers.

The nippers are not a precision tool. With a little practice you can get close to what you want. Luckily doing mosaics breaking a couple of squares is relatively cheap. If you have a weird corner cut on a tile or a circular cut and you can get it done in 2-3 tries the great job.

#3 For straight cuts you use a wheel scorer. The picture I have below is a double sided. Often you will just see a single wheel. Just like cutting drywall it is most important to get a good score on the glass side. With a little muscle you should only have to roll an area 2-3 times.

#4 Once you have a line on the top of your tile we are going to move it over to the tile cutter. You are going to place this faced up with the edge right on the cut line. This takes practice and is a little different with each cutter. You will gently (do it to hard and cut may not be clean) bounce the lever to put pressure on the tile. If it is lined up right eventually it will break clean on your scored line. If your cut isn't clean then you may need to score a little deeper or line it up better on your cutter. Also do one tile at a time. I know it seems a PITA but you can do 2-3 tiles in a minute once you get it down. Also do your cuts before you start applying thinset or have a helper.