Is there a name for deformation-based maps?

This is called a distance cartogram. Some add-ons for ArcGIS have been produced to create area cartograms. Mark Newman provides free GIS-independent software to implement his area cartogram method. Methods of creating distance cartograms are known in statistics as multidimensional scaling (MDS). I do not know of any GIS software that implements MDS, but it's available in full-featured statistical packages like Stata (commercial) and R (free).

Anamorphic map or even cartographic anamorphosis are also used.

Scapetoad does it.

Scapetoad produces area cartograms, not distance cartograms. You can produce distance cartograms with Darcy You can download the software (Java program) : https://sourceforge.net/projects/jdarcy/

Before using Darcy you have to transform the distance matrix in coordinates (multi dimensional scaling). In R the simple function for MDS is cmdscale. Now, Darcy implements this R function.