Could you tell me now what is the sword with rings in this clip?

I would say it is just a customised Dao

It appears to be either a Baquashendao, Liuyuedao, or a Yanlingdao.

With those rings and ornamentation on it, it looks like a Jiuhuandao from @Espiritus_Corvus’ link.

I guessing that the rings have some purpose other than ornamentation. Is this correct?

It is, indeed, a jiuhuandao (which is typically translated as “nine ring broadsword” or “nine ring saber”). The rings are not found on military weapons and appear to have been added to make demonstrations more entertaining. In that sense, they started out as purely decorative. Nevertheless, practitioners have proposed various practical uses for them.

Some claims I have seen made on behalf of the rings are: they can be used to catch spear heads (which might help you disarm the attacker), they absorb some of the force when blocking with the back of the blade (it’s easier to replace a ring than a sword spine), and they can distract an opponent (though only one who is already a bit unfocused).

As the film clip shows, however, they can also be a bit of a liability. It’s a lot easier to grab the rings than it is to get a solid grip on a bare sword spine (though grabbing the rings comes with risks of its own).