Keith, you are quite correct. The "daito" of Daito Ryu is 大東 (great south) while the the other is 大刀 (big sword). 大刀 is a pretty uncommon usage. Usually they go with 太刀(big sword) or just katana 刀. I will add however that usage rules in kanji are a mess, and I don't ever expect to fully understand them.
Actually daito is used rather often in the world of those interested in the history of the Japanese sword and/or collecting of nihonto. If it's any more clear, daito is a generic term for long sword. Tachi refer to the earlier swords worn edge down slung from hangers. The "daito" of this period are called tachi in specific because of both their shapes and also their functional mode of wearing. Tachi will have the smith's signature on the "other" side of the nakago as compared to katana of later history (worn edge up). Many tachi were shortened from their originally very long and graceful shapes in to shorter versions more amenable to the "new style" of wear. So given that context collectors will tend to refer to any blade longer than a wakizashi as a daito especially if they're trying to avoid making any assertion as to more specifics about its provenance. So a marvelous suriage'd Go Yoshihiro I was quite honored to have spent time studying a few years ago was a tachi shortened at some point and remounted as an uchi katana. It is a daito as it was still longer than a wakizashi. Is it a tachi still? Is it a katana? It depends on how you look at it. It was originally a tachi suriage'd (shortened) into a size amenable to wearing as a katana. But still a daito.
As I said it is also quite commonly used when you have sword either forged as a set with a long and short blade *or* you have a long and short sword *mounted* to go together as a matched set. Both together are considered a daisho with the daito being the long sword, the shoto being the short one. The daito could be a katana or tachi. The shoto could be a wakizashi or tanto.
So in the collecting/studying nihonto crowd daito is a rather generic term simply meaning "long sword". There are more specific words for specific examples of long swords.
Yeah, it's that kanji thing again. And not to mention a greater than 1000 year history with all sorts of terms going in and out of vogue. You can go nuts keeping track of the various terms that often refer the bloody near the exact same thing.
Attached is one of my favorite pictures ever. There is a kanji that looks *similar* to the kanji on his shirt that is pronounced the same that means "samurai". Unfortunately the kanji he has actually means hemorrhoid.