Karen started the whole ball rolling with Don't Shoot the Dog, but that's not a dog-training book. Rather, it is a popular presentation of the essentials of operant conditioning.
Karen and Gary Wilkes introduced operant ("clicker") training to dog trainers with a series of seminars in 1992 and 1993. Gary located the box clickers and both he and Karen distributed them at the first clicker training seminar, in San Francisco in May of 1992. I became a convert at their Anaheim seminar in March 1993. In 1994, Karen and Gary split up as a team due to essential philosophical differences, principally over Gary's willingness to use aversive techniques—his "bonker."
So, sometime around the middle of 1993 there was a very small group of people—Sue Garrett, Sue Ailsby, Corally Burmaster, Gary Wilkes, Debi Davis, Kathleen Weaver, Steve White, myself, and several others sprinkled here and there—participating in the first clicker training online discussion group, click-l, started by Kathleen Weaver, a Texas trainer. We were all more or less starting from scratch and talking about what we were doing, our mistakes, and our successes. We had all the fuzzy-headed discussions about behaviorist terminology, and more. Around the time that I was articulating how I was teaching signals, Karen wrote to me and said, "You are working on a book with this stuff, you know."
My book, Clicker Training for Obedience, was published in 1998, a month or so after Alexandra Kurland's first book, and together they were Sunshine Books' first formal entry into the world of clicker training for dogs and horses. Since then, of course, there have been myriad articles, books, videos, etc. I wouldn't even want to try to start to list all the publications and lists that have been founded in the intervening years. Bob and Mouse Bailey had a lot to do with it in the late 1990s and early 2000s when they started their chicken camps; we did several "chicken/dog" seminars during that time, including two or three at Terry Ryan's Legacy Camp. And speaking of Terry, Jean Donaldson, and others in that vein, while they were not clicker trainers as such, they promoted a positive philosophy of dog training and really did work with operant principles even though they didn't always articulate them that way. In that camp we also include trainers like Patty Ruzzo and Dawn Jecs.
So, "Who started clicker training for dogs?"
We all did. And we're all still doing it.