Most likely your mesh contains incorrectly oriented triangles. Older Slicer versions by default did not show backface of triangles. You can enable display of both sides in Models module / Display / 3D Display / Visible Sides → All. Current Slicer versions show both sides by default and contain many other fixes, so most likely they will show the full mesh by default.
A .vtk file can contain an image, surface mesh, or volumetric mesh. If Slicer only offers to save a model in .vtk and .vtu (unstructured grid) format then it means that the model contains a volumetric mesh, which cannot be saved as a surface mesh (.stl, .obj, .ply, … formats).
You can convert your volumetric mesh to a surface mesh by copy-pasting this code snippet into the Python console (replace Model
by the name of your model node):