While someone somewhere has surely written a nice clean, unified explanation on associating
favicons, Apple Touch icons, and site thumbnail images with WordPress 'blogs, I've not seen it; and, when it comes to supporting site thumbnail images in WordPress themes, I have seen an awful lot of discussion that is unclear, and PHP code that is inflexible, breakable, or already broken. Perhaps I'm now about to contribute to the problem, but I'm here going to make my own attempt at discussion.
[This entry was written at the suggestion of the Woman of Interest, but she's really not to blame for how it turned-out.]
I think that a great many people are just going to want code that they can drop into a theme, and a practical explanation of how to use that code, so I'll provide such code, with an explanation of how to use it.
A few hardy or lost souls are going to want to know what motivated that code, how it actually does what it does, or what I see might be done differently. (Some of them might do a rather better job than have I.) For those readers, I have prepared a very boring discussion.