Interior of discretized region

It can be done like this:

Needs["NDSolve`FEM`"]
mesh = ToElementMesh[dom];

boundaryIncidents = Union@Flatten@ElementIncidents[mesh["BoundaryElements"]];
allIncidents = Union@Flatten@ElementIncidents[mesh["MeshElements"]];
interiorIncidents = Complement[allIncidents, boundaryIncidents];

allCoordinates = mesh["Coordinates"];
boundaryCoordinates = allCoordinates[[boundaryIncidents]];
interiorCoordinates = allCoordinates[[interiorIncidents]];

Show[
 mesh["Wireframe"],
 Graphics[{
   PointSize[Large],
   Red, Point[boundaryCoordinates],
   Blue, Point[interiorCoordinates]
   }]
 ]

You can think of incidents as the indices of coordinates in the list mesh["Coordinates"]. The idea is the same as that of GraphicsComplex if you have ever used that. mesh["BoundaryElements"] gives you all the line elements that make up the boundary described using incidents. mesh["MeshElements"] gives you all the triangles that make up the mesh described using the same incidents. All we have to do to get the coordinates that are not on the boundary is take the incidents of the full mesh and remove the incidents that are used in the boundary elements.

Here is another approach:

mesh = ToElementMesh[dom, "MeshOrder" -> 2];
bmesh = ToBoundaryMesh[mesh, "MeshOrder" -> 2];
boundary = bmesh["Coordinates"];
interior = Complement[mesh["Coordinates"], boundary];

I specify "MeshOrder" just to make sure it will be the same in the mesh and the boundary mesh. I noticed that by default the boundary mesh gets mesh order 1 and the full mesh gets mesh order 2.

Another option, provided by user21:

mesh = ToElementMesh[dom, "MeshOrder" -> 2];
coords = mesh["Coordinates"]; 
boundaryIDs = Flatten[ElementIncidents[mesh["PointElements"]]];
interiorIDs = Complement[Range[Length[coords]], boundaryIDs];

Look this post

region = DiscretizeRegion[dom];
Show[region, Graphics[{Red, PointSize[.02], 
   MeshPrimitives[DiscretizeRegion[dom], {0, "Interior"}]}]]

Using mesh region functions:

reg = DiscretizeRegion[dom];
boundary = RegionBoundary[reg];
coords = MeshCoordinates[reg];

pointindex = 
  Pick[Range[Length[coords]], RegionMember[boundary, coords], False];

MeshRegion[reg, MeshCellStyle -> {{0, pointindex} -> Red}]

If you want to get coordinates of interior points:

coords[[pointindex]]

or you can do from beginning:

Pick[coords, RegionMember[boundary, coords], False]