Is there a term for mapping between two arbitrary quadrilaterals without using homography / perspective projection? When warping the contents of an arbitrary quadrilateral to another by extrapolating equal points along both the "lengths" and "widths", thus ignoring all perspective effects / foreshortening, we are not using a "homography" or "affine" mapping.

Is there a correct name to describe the type of mapping we are doing?
Yes, this is called bilinear interpolation or a bilinear mapping. The term "bilinear" crops up more commonly in the context of image resizing or resampling, but it's the same basic idea for transforming quadrilaterals—it's just that the interpolation is applied to the vertex positions of the quad, instead of the colors of pixels in an image.

One paper that briefly discusses bilinear mappings of quadrilaterals is Fundamentals of Texture Mapping and Image Warping by Paul Heckbert; see section 2.2.2.