Tony, the important thing to remember is that all the notes have half their normal value, and that the song is felt in a 2 beat tempo accent on the second beat. In 4/4 time the accent would be on the 2nd and 4th beat.
Yes, the keyboard beats two times in each measure. Normally, because of the speed and beat/accent feel it is easier to count 12 12 then 1234.
In learning a song I have always counted 1234. After I have learned the song counting was not necessary and my concentration was on 12 12 with the accent on 2.
As far as the keyboard goes, one beat equals two written quarter notes.
You have the same problem in 6/8 timing. Do I count 123456 or 1 2, 3 eith notes equal one beat.
Again it is the feel of the rhythm. Fast 1 2 Slow 123456. Slow is typical of a waltz written in ¾ time. Fast 123456 with the accent on 1 and 4 you begin the get a triplet feel 123 456 123 456 with accents on 1 and 4 – Tarantella – Irish Jig are examples.
Hope this helps, John C.
PS, I had been playing professionally for a few years before I understood cut time, I just treated it as a fast 1234 1234 but I did understand the feel of cut time.