This different view on things might even be the reason why Arrangers are much much more popular in europe then in the US.. To many of us an arranger is a mini organ, to you guys its just an arranger. Even now in euopre most OMBs have still 2 keys on stage, from which atleast one is an arranger, which again shows how deeply double manuall organs are routed in our minds.
I believe the reason I have embraced arrangers so thoroughly was because I was an Electone organ teacher for several years, being a graduate of the Yamaha Electone School...I was teaching on, first, a Yamaha Electone C-30 and then an E-75.
At home I had an Electone D-85 (triple manual) and then two C-605's, the later one a very rare factory portablized model.

These were analog instruments, based mainly on the Yamaha CS-80 synthesizer technology, and they had the early auto-accompaniment (ABC...Auto Bass Chord), although, at that time, I was primarily using pedals.
Owning, and teaching on, these instruments did give me another perspective on organs...Hammond was also branching out into transistorized organs with auto-accompaniment, but I still kept the old B-3 for band use.
Prior to these instruments, and probably also popular in Europe, were the various Farfisa
combo organs I used, although my very first combo organ was a second hand Howard Baldwin. These instruments (the Farfisa) did not have auto-accompaniment, but did allow using bass pedals. I ran all these combo organs, the last Farfisa being the VIP-233 model, through a Leslie 147RV, which gave me a fairly Hammondy-like sound (as you say, we North Americans associated "organ" with "Hammond"), and eventually I finally picked up a second hand B-3 and added a second Leslie 147RV. The genuine Hammond sound was what was wanted for the R&B/Rock bands I was working with at the time.
However, the analog Electone were the basis of the early PS-series Portatones and, of course, we sold them at the store where I taught, and, eventually, I ended up using, and teaching on, a PS-55 and later a PS-6100 (which was digital FM based), so again, this is why I have embraced arranger keyboards so enthusiastically, compared to many of my other keyboard playing buddies who usually stayed with the Hammond and subsequently went to clonewheels. At this point, I had become very comfortable with adapting to auto-accompaniment style chording.
My first OMB rigs were usually an Arranger and another keyboard, either a synth or portable piano, so the dual manual organ influence stayed with me for quite some time. Nowadays, the Tyros4 will cover all my needs very well, although, if I do gig again, I'm considering adding a second keyboard, the type of which will depend on what kind of music I'll be focusing on.
Great topic, by the way...it's interesting to see how our cultural backgrounds have affected our choice of instruments, and, music in general.
Ian