Why Walk Off?

I'd gotten so tired of every baseball game that ends with the home team taking the lead in the bottom of the ninth being referred to as a "walk off." I started thinking, "Why walk off"? Isn't every game a walk off? Either the home team wins by getting the away team out in the bottom of the ninth (and walk off the field), the home team wins by a home run or any run scored and the away team walks off the field (while the home team's player may actually be running to the plate to score the run), or the home team fails to win game in bottom of ninth and again the away team walks off the field (to hand shakes and high fives).
So I looked it up (thank you wikipedia. Seriously, who takes the time to post an entry to "Walk Off Home Run"?) It looks like the phrase came from Dennis Eckersley who called game ending home runs, "Walk off pieces", because the pitcher would "walk off" the mound with his head hung in shame. Now that makes sense.