I moved onto a lumberyard improvement (to my parties' right) located outside the city walls (shown above my party) and captured the city.  The troops stationed 'inside' the city defended the lumberyard.

Suggest that we be able to pillage the improvement but city capture only occurs when attacking the city 'proper' (ie -- the 'walled' part).  Troops could be stationed on the improvement to defend it, but they'd be separate from the troops 'in' the town.

This would make improvements 'outside' the city more vulnerable.  It would give defenders a reason to sally forth, and attackers a way to hurt the defender without attacking the city proper -- making raiding a viable strategy.