DEM (Digital Elevation Model), DTM (Digital Terrain Model) and DSM (Digital Surface Model) are often use interchangeably and it can be difficult to discern them between each other, especially if the provider considers them the same. You should be able to create a DTM by using a detailed DEM though. Have a look on these sites and see if you can find something usable, see:
2. What is the spatial resolution of the DTM required for your project.
For free you can get AW3D30 - a very good model (one sigma approx. 0.8m), and a much more popular SRTM somehow less accurate (one sigma approx. 2m). Both are at 1" resolution and both are DSMs.