I can't give you a definitive answer because I suspect there are others I don't know about.
Here are the ones I do know of:
Tenkara - fly fishing on mountain streams (and "main streams" ie. rivers in the mountains)
Keiryu - bait fishing on mountain streams (and main streams) also referred to a myaku tsuri (Pulse fishing, similar to French au toc, tight line fishing feeling for the "tap tap" indicating a hit)
Seiryu - bait fishing on "clear streams" - streams that are out of the mountains, but generally not major rivers, using floats, targeting smaller fish (chubs, etc). They hold competitions for this.
Hera - bait fishing for crucian carp (MUCH smaller than our common carp) in lakes, with floats
Tanago - bait fishing, largely confined to lakes and ditches for bitterling (max. about 4"), with floats
Mebaru - bait fishing for salt water "rock fish" from shore

I know some people fish for a species of shrimp called "kleptomaniac shrimp" using bait, but I don't know what the style of fishing is called.

There are fixed line rods designed for carp (common carp) fishing and others designed for salmon fishing (bait fishing in both cases).

To my knowledge, there is no fixed line fly fishing done in Japan in lakes for bluegills and bass, etc, even though they have bluegills and bass. There is some fixed line fly fishing for the fish seiryu fishermen seek, but I don't think it is common and I don't know whether it would be called tenkara because it is fly fishing or seiryu because the streams are not mountain streams. Some tenkara fishermen use seiryu and keiryu rods when tenkara fishing.