Um... [b]<<Slant>>[/b] Is a skill that doesn't utilise a skill slot
The skill slot utilised would be [b]<<One handed straight sword>>[/b]
So that you can use different sword skills. It wouldn't make sense to have to equip all of those sword skills.
Kirito had more than 12 sword skills available, and could use them all. He had 12 skill slots at level 96. Doesn't make sense. Sword skills are like a skill subset.

And I am unsure if there is a way to check which sword skills you have technically unlocked or can use. I am unaware how that works, if they simply activate it and realise or if there is a list or something with requirements to unlock

Edit:
You don't learn most skills just like that. Some skills are available by default and require to be equipped in the skill slots to be used. Others require to be unlocked, by quests or other special means.