When does a knife stop being a knife and start being a sword?

The distinction is more or less subjective as far as size goes. Form factors for edged weapons are all over the place. There are examples of single-edged swords along with double-edged swords in just about any culture and time period you care to name. The line between short sword and long knife gets pretty blurry in some cases. There’s just so much variation in form and use that you can’t really make a foolproof definition that holds up in all cases. In general, a blade that’s over about 18 in. (45 cm), designed exclusively for fighting with no utilitarian use would probably be called a sword, rather than a dagger or knife.

Ultimately, the terminology is up to the people who made and used it. After all, a cinquedea was usually right in the middle of the length range for a wakizashi, and the latter was most definitely classified as a short sword, but a cinquedea was considered a dagger by the Italians and Spanish. Same thing with seax and dirks; those were considered long knives, even though some of the longer ones were getting into the usual range of lengths for short swords.

Something Kinthalis mentioned earlier, about weight distribution, is relevant I think. Knives and swords generally feature different weight distributions. Swords are designed so that the center of percussion is at a good spot for a cutting strike, and the vibration from hitting or being hit by something doesn’t hurt your hand, while at the same time not being so blade-heavy that it’s unwieldy. This kind of weight distribution is not usually found in knives, even large ones designed for fighting, but is generally found even in swords that are designed for thrusting.

Rapiers or even smallswords would never be classified as a long stiletto or something similar, for all that they’re almost exclusively thrusting weapons. They are evolutions in the design of swords, not grossly elongated knives. An estoc also had no cutting edge, but no one would say that it wasn’t a sword.

Sai are blunt weapons, similar to a baton, and are technically Okinawan, not Japanese. The Ryukyu islands were an independent kingdom for most of their history, became a Chinese territory for a couple of hundred years, paid tribute to the shogun as well as the Chinese empire from the 1600s, and didn’t get officially annexed until the Meiji era in the late 1800s. Not that most people make that distinction any more, but karate and the associated weapons are pretty much unrelated to anything in the main Japanese culture.

The Wikipedia article is pretty decent, though the link to a similar Japanese weapon, the jutte, is lacking.