Re: When did the word "Aiki" first appear?
Regarding the phrase "aiki" or "aiki no jutsu" and it‘s introduction to the vocabulary of usage in martial arts, it will certainly take someone well versed in the Japanese language to do the research.
The earliest English reference I am aware of is the 1912 book,"The Fighting spirit of Japan" by E.J. Harrison where "aiki" is mentioned several times, and associated with the Shinden Isshin ryu.
In the latest DVD by Kondo Katsuyuki, current Headmaster of the main-line Daito-ryu, he mentions the phrase "aiki" is written about in a book by Sato (kyoju-dairi) called "Japanese Spirit" which was written a few years after Sato had begun his training in Daito ryu (1902.)
There is a Japanese publication written in 1892 (and later reprints) regarding "Aiki" and various aspects of training in this esoteric martial art.
As to an even earlier reference that can be documented of "aiki no jutsu", it pops up in a semi-novelization of the famous swordsman Miyamoto Musashi in 1887. The kanji for "ki" is an out-dated one - 氣.
So, the term "aiki" and its understanding in a martial way were already mainstreamed by the late 1880's.
On a personal note, I am not a believer that Onisaburo Deguchi convinced Takeda Sokaku or Ueshiba Morihei to add the term "Aiki to the Daito-ryu jujutsu name. Further research is needed.