The Ki Society is a centralized organization. That means the teaching methods are the same in Japan as they are in the US or elsewhere. Therefore they don't take into account the differences in culture. If you tell a Japanese student, "Extend Ki." they might have an idea of what you're talking about, since ki is a fairly common word in the Japanese language. If you tell an American student, "Extend Ki.", they will scratch their head and ask, "What's a Key?" since the word is foreign to the English language.
Interestingly, Japanese students are told 氣を出す ki wo dasu
is one of those ubiquitous, heavily used words in Japanese (possibly used in more contexts than ki!), but the core meaning is one of "put out", putting something that is inside outside. A look here
can give you idea of the many contexts it's used in. To my ear, dasu
seems to have a much broader meaning than "extend", but I imagine those in the Ki Society tried to find a proper English term for the feeling they wanted to evoke.
Also, I find it interesting that the Ki Society uses the 氣 character, to distinguish it's ki from the 気 ki
used in everyday Japanese speech. In contrast, Aikikai uses 氣 only in calligraphy, and uses 気 in all other situations, including 合気道 aikidou
and 合気会 Aikikai
Incidently, when Morihiro Saito refers to ki
in his Takemusu Aikido series (in, IIRC, his Traditional Aikido series), he says to 「十分に指を開いて気力を充実させる」, "Fully open the fingers and fill them with ki power."