The main weapon of the samurai at the time was the bow, used either on horseback or on foot.

The accompanying hand-to-hand weapon was, as you note, the tachi. A tanto would have also usually been carried.

As for the difference between the tachi and the katana, the defining difference is the mounts, specifically the way that the scabbard is intended to be worn. The tachi is hung from the belt, edge down, and the katana is worn through the belt, blade up. A blade made as a tachi blade can be distinguished from a blade made as a katana blade if it is signed; they are signed on the "outside" side of the tang. That is, on the [left side of the tang for a katana, and the right for a tachi] (https://commons.wikimedia.org/wiki/File:Katana_mei_and_tachi_mei.jpg). There are tachi blades out there that were remounted as katana. This difference in mountings is due to the tachi being designed for cavalry use, and the katana for wear on foot. Cavalry swords often being longer than infantry swords, it is unsurprising to see that tachi blades were often longer, with many Kamakura examples having between 80cm and 90cm of blade, which would be exceptionally long for katana blades. For many examples of Kamakura blades, see Markus Sesko, "The difficulties in classifying Kamakura blade shapes", https://markussesko.com/2013/02/24/the-difficulties-of-classifying-kamakura-blade-shapes/

There are Kamakura Period katana-signed blades, so the katana was in use. However, it was not as common as it was later.

Shorter swords were in use (and would have usually been called kodachi). Most of the National Treasure blades from the Kamakura are tachi, katana, or tanto; there is only one such kodachi.

The most common polearm appears to have been the naginata. The Kamakura naginata was typically much longer-bladed than an Edo naginata, often closer to a nagamaki in proportions than to an Edo naginata. There are surviving spears from the Kamakura Period, but they're quite rare, and artwork shows naginata in large numbers, and very few spears.