Without reference to any book, I seem to remember that a "sensitive" altimeter has the kohlsman window and therefore can be set to the local barometric pressure. i.e. it is sensitive and thereby can be used for IFR flight.
A pressure altimeter doesn't necessarily have to be sensitive. It could be locked in to some number (usually 29.92 or standard) and only show the differences in altitude subject to atmospheric changes. Ergo, you start at a sea level airport but the altimeter shows 200ft. You climb up to an altitude of say 2000ft over the airport. Well, you better be at 2,200 keeping the same differential in the air as on the ground. i.e. a VFR only airplane doesn't have to be "sensitive" because you can see and guestimate the obstacles and mountains.
Pressure Altimeter is another name for a pneumatic altimeter. A pneumatic altimeter is a form of aneroid barometer whose dial is calibrated in feet or meters, rather than units of pressure. A pneumatic altimeter has a barometric scale that can be adjusted to any desired reference pressure, and the pointers of the instrument measure the difference between the existing pressure and this reference pressure in terms of feet or meters.
When the local altimeter setting is set on the barometric scale, the altimeter shows indicated altitude. When standard sea level pressure of 29.92 inches of mercury or 1013.2 millibars is set on the barometric scale, the altimeter shows pressure altitude.
A sensitive altimeter is a form of multipointer pneumatic altimeter with an adjustable barometric scale that allows the reference pressure to be set to any desired level. Sensitive altimeters have a high degree of mechanical amplification in their movement. The small amount of movement of the expanding bellows inside the instrument is multiplied enough to rotate one of the pointers completely around the dial for a pressure change equivalent to only one thousand feet of altitude. This is approximately one inch of mercury.
What does all this mean? All altimeters are pneumatic altimeters; pressure altimeters. Not all altimeters are sensitive altimeters. Some altimeters are much less sensitive. One that leaps to mind is the standard alti found in the origional piper cub, and many other aircraft of that era. The needle could be set for field elevation, but no barometric setting was available.
When I skydive, I wear two altimeters; one on my wrist, and one on my chest. I zero them to field elevation (showing ground level as 0 on the altimeter) before we takeoff. These altimeters are pressure/pneumatic altimeters, and I can rotate the pointer to read zero. However, there's no barometric compensation; these are not considered sensitive altimeters.
The cabin pressure altitude indicator in the airplane you fly is a pressure altimeter, but not a sensitive altimeter.