it's called what ever the hell your preference is.
Nice way of putting it.:scowl:
holuoch, knowing your budget would help us make some recommendations.
Neck through or bolt on are really your personal preference. The church I go to plays a lot of urban gospel stuff, even though we are a small church in a whitebread town in Arkansas. Our Pastor is also our worship leader, and a formidable pianist. He likes to do hymns with a blues feel, playing harmonica while comping on piano, and all of our more modern stuff is very urban influenced. I have used bolt on, hybrid, and neck through, and all have worked equally well IMO.
A good starting place is a used Peavey Cirrus. They won't break the bank, you can usually get one for around ~$900 - $1100 in our Classifieds or on Ebay, depending on the woods.
As prices go up, things like Roscoe, Zon, Nordstrand, Sadowsky, MTD, Dingwall, Ken Smith, and Fodera work very well, but there are a lot of other brands I haven't mentioned that do the job.
I primarily play a Nordstrand SC5 (hybrid), but also play a Cirrus (neck through), a Zon (fretless bolt on) and Lakland Skyline (bolt on), and they all do the job well.