Appreciate all the responses. Meanwhile, I did some web searching, and found the following from what appear to be reputable sources.
The reason glyphosate becomes inert once on the ground is from bacterial action. That's why there is no "defuser" you can add to it for deactivation, and why if you apply it over mulch it will not automatically degrade.
Common herbicides that are much longer-lasting in sprayers, especially after sitting for a day, are those with 2,4-D and dicamba. You can get some of it out with water, commercial tank cleaners, household ammonia left overnight, detergents, the cleaner trisodium phosphate OR (better sit down) glyphosate itself. (About ammonia, you want a 1% solution of 3% ammonia, and keep it away from chlorine/bleach unless your insurance is current.)
About killing roses from a contaminated sprayer, if I did that in the boss's rose garden, she'd cut off my wand, so thanks for the warning. Are you sure you didn't once have 2,4-D and/or dicamba in the sprayer?