And fiber cement is quite a bit thicker and denser than vinyl so I would have a much thicker looking wall if I had fiber cement with 2 or more inches of foamboard underneath it, right?
Two things going on there. Fiber cement board is a little more than a 1/4" thick, so in that respect it is "thicker" than vinyl siding, but vinyl siding is made to imitate wood planks which themselves are much thicker than fiber cement. So, in the long run, you get a "thicker" look from the reveals or shadows for both wood plank and vinyl. HardiePlank has thin, crisp lines.
Vinyl requires no painting, whereas fiber cement must be painted, although it seems to take paint better and be longer lasting than wood. In my area of the Pacific Northwest, vinyl siding seems to grow mold and mildew, so instead of painting, you need to wash it periodically.
Back to "thicker" walls..... Adding some thickness to the walls, as you might with foamboard, doesn't really change the overall look of the house from any distance, but up close, you might see that the window sills and door penetrations are a little deeper. You would have to be looking for it to notice. Adding a couple inches to a 4" or 6" wall still doesn't make it very thick. With ICF, for example, your walls are 12" thick and it has to be pointed out to most people for them to notice.