Why the rug waits until he is dry
A rug is insulation, and insulation over a wet coat seals the wet in. The horse under a rug put on too soon stays damp for hours, his back muscles cool slowly under a clammy layer, and the friction of rug on wet hair rubs far more readily than on a dry coat. The one exception is a purpose-made wicking cooler, which is designed to pull moisture away from the coat and let it evaporate; that is a drying tool, not the rug he stands in afterwards. So the rule for both situations in this question, the deliberate bath and the ride that ended in rain, is the same: dry first, rug second, and judge dryness by the skin, not by the surface of the coat.
After a bath: scrape, cooler, walk
Run the sweat scraper over him firmly and repeatedly, because most of the water in a soaked coat leaves mechanically rather than by evaporation. Put the cooler on, making sure it sits up on the withers and is fastened so it does not slide back as he moves. Then walk him, in the sun if there is any, in hand or on a long rein, rather than letting him stand in a draught shivering the water off. A horse who will graze in hand can dry while he eats, which suits a hot day. Feel into the coat with your fingertips as you go: the barrel dries first, and the honest test is the skin under the mane, behind the elbows, along the belly line and between the hind legs, which can still be wet a quarter of an hour after he looks dry.
After a wet ride: the legs are the priority
Untack first and deal with the saddle area: sweat and rain under the saddle should be sponged or rinsed off so salt does not dry in the coat. Then turn to the legs, which have usually come home plastered. Hose or bucket-wash the mud off, and then dry the legs properly rather than leaving them to air: towel them, paying attention to the heels and the backs of the pasterns, and feel for any cut, swelling or heat the mud was hiding. Legs left wet and muddy in the stable day after day are the classic route to cracked, sore heels, so on persistently wet weeks some owners dry the legs and apply stable bandages or let the horse stand on deep dry bedding while they dry. His back can be drying under a cooler while you work through the legs, so the two jobs overlap.
Rugging the dry horse, by feel not habit
Once he is dry to the skin, choose the rug by the horse in front of you: clipped or unclipped, fat or fine, stabled or turned out, and how the night is actually shaping up rather than what the month usually does. Slide a hand under the rug at the shoulder after ten minutes; it should feel comfortably warm, not sweaty and not cold. Fasten chest straps first, cross the surcingles with a hand's width between strap and belly, and link the leg straps so they cannot ride up. A rug put on a genuinely dry horse in the right weight is a comfort; the same rug an hour earlier was a cold compress.
Make it the house routine
Horses get rugged half-dry most often not through ignorance but through different people finishing up differently: one rider waits, another is in a hurry, a helper assumes the first checked. Writing the sequence where everyone sees it ends the guessing — scrape, cooler on, walk until the skin is dry, legs washed and dried, then rug by condition and weather. Give relatives you pick access to his profile via KinsLe Family Sharing, so chosen family members can open the profile of the horse who must be dry to the skin before his rug goes on.