I don't think it says so much about the way people remember things as it does about people exaggerating something when they think no one will ever find out the truth. He probably wished he had inscribed the watch with those words and so that is the way he told it. He probably even came to believe it himself after many years of telling it that way.
At least the basic story was correct. Another thing, he may have told the family what he really wrote in the watch and the family added to it over several generations.
People tend to view the past through the lens of things that happened or were found out later. I don't think any conscious desire to deceive was at work. Because the war ended slavery the watch repairman convinced himself that he knew at the time that the beginning of the war meant the end of slavery and inscribed the watch that way. Maybe it was like the game of "telephone" where a message gets altered by being repeated from person to person, but in this case, it was one person over a period of half a century or so.
I shouldn't have implied that everybody remembers things that way. But look at what Obama said about his uncle liberating Auschwitz. He remembered something that his uncle said about a concentration camp and Auschwitz was the concentration camp he'd heard most about so he put the two together, even though it was actually the Soviets who were active on that front. Oral traditions tend to change in that way if people don't have fixed text sources to turn to.