Same thing later in the tech tree: I just did the research that unlocks caravels, another type of transport boat, and those too don't appear in my list of units.

Just go with your land units into the water, the ship will spawn automaticly.

It's confusing and VERY poorly worded, but all "Transports" are upgrades to the basic embarking movements

Yeah, just move the land unit into the water and the boat spawns.  Big issue then is it gets only two movement points.  If you need to cross deep water greater than 4 tiles away, your unit dies.  There is this "boarded" mechanic which I thought was to load a unit onto a ship with longer range but can't figure out how to get the unit "boarded".

Thank you very much for your answers! And yeah, it's definitely not intuitive at all... I finished the game with a technological victory while staying stranded on my starting continent. But this will definitely change things for the next game!