Just as your posted video shows, it isn't just aikido that use the term irimi. I wouldn't get too tied up with aikido's term because their irimi is quite specific and involves a series of movements.
"Irimi" just means "entering" and goes back to both 'ti'/'tode' and Japanese arts such as kenjutsu and jujutsu. Tenkan is really just irimi + a turn. Again, in aikido the concept is more elaborate (I'm not being critical, just that this would be a very crude description of aikido's sequence of moves and their concept).
Your posted video shows 'ura irimi' which is an application of the opening move of naihanchi shodan.
On the other hand saifa uses standard 'irmi' as do other Okinawan kata.
Both occur in the Chinese arts, particularly the internal arts but also some Shaolin. Here's me in around 2000 demonstrating irimi from Hong Yi Xiang's form "Shaolin Peng":
I actually don't know the Chinese term (they aren't big on naming individual applications of forms), but people like Tim Cartmell are experts at it (he is a bagua/taiji practitioner who is also an experienced full-contact fighter in the Orient).
Have a look at our touxing forms - a collection of qin-na 'irimi' of the kind people like Cartmell utilise.http://www.youtube.com/watch?v=iiUaVspH3TAhttp://www.youtube.com/watch?v=B56GG6bCdVU