Okay, as best I can tell, MastoWeb uses OpenGraph information to construct their preview cards, which does include the post's Featured Image (as OpenGraph thumbnail) as long as it's big (1920x1920 works, 300x300 does not).
Ivory, by contrast, either sources OpenGraph and then doesn't use half of it or uses Twitter Card data. (It doesn't display website name.) It, too, wants a BIG Thumbnail or else it will post no preview card whatsoever.