Hi Drakanyst,
Well, where to start? First off, as you have put this in the Japanese Martial Arts section, let's look at that. The examples shown are examples of Okinawan methods, which are different to Japanese methods. Okinawan usage of a Bo tend more towards holding the hands each one third of the way up the staff, and keeping them there. Additionally, due to the Chinese influence, one of the primary methods to teach it is a solo training drill, refered to as a kata (which means "shape", or "form", implying that the method keeps the same form as it is repeated, without deviation). They then can be examined by looking at the application of the solo movements against an attacking partner, which is what is shown in Bill's clip above (this is what is refered to as "Bunkai", which basically means "application"). The other one (from Harlan) is a stunt group, rather than an actual martial system, for the record.
For what you are looking for (Bo versus another weapon), that is done in many of the Japanese systems, in their form of kata (which, particularly in the older systems, is a paired exercise, with one partner being the "reciever" - the person having the technique done on them - and the other being the "performer"). Many will utilise the Bo against a Sword, others against weapons such as Naginata, or another Bo, and so on, depending on the system itself. It should be noted that a Japanese method of using a Bo is fairly different to an Okinawan usage, with the hands moving from one end of the Bo to the other freely, taking more advantage of the range of the weapon. Some examples are as follows:
Tenshin Hyoho Kukishin Ryu
Kukishin Ryu Bojutsu (as taught in the Genbukan)
Tenshinsho Den Katori Shinto Ryu Omote no Bo
A rare one... Katori Shinto Ryu Gogyo no Bo (Bo versus Bo)
Yagyu Shingan Ryu Taijutsu Bojutsu
It really should be noted, though, that if you are looking at Okinawan methods, then Okinawan methods tend to not have these type of kata.