I am certainly lacking knowledge of bush or vine honeysuckle that would be a none invasive type. Especially one that would thrive in your climate zone.

{{gwi:8337}}

Sorry I don't know anything about bush honeysuckles, but below is just one site I found by Googling "bush honeysuckle." If you click on the links at the bottom, maybe the pictures will help you i.d. your neighbor's shrub.

You mentioned your neighbors cut theirs back each year, though, and to me it seems a shame to fight the plant's natural growth like that. How big do you want it to get? I would recommend clethra alnifolia, which would stay smaller ('hummingbird' I think is the smallest)and could take your zone and soil conditions. Not sure if they attract hummingbirds, but 'ruby spice' is a nice pink color. Some varieties also turn a pretty shade of yellow in fall. Just a thought.
'Mare

Here is a link that might be useful: invasive honeysuckles

I have a Honeysuckle called Woodbine Honeysuckle. It smells plenty fragrant but not that strong as that of the regular one. After flowering, there is a cluster of bright red berries. It can stand -20 to -10 degree F for Zone 5-9.

The instruction indicates to prune annually to control the size. Make a raised bed for planting. Most regular plants hate wet feet.