The term spur gear usually refers to a straight (parallel to the axis of rotation) cut gear regardless of tooth geometry. This differentiates them from helical gears which have teeth cut at an angle to the axis (other than zero).
Cycloidal gears are used in clocks and perhaps because of some connection to his fascination with pocket watches Ford used cycloidal gear teeth in the Model T transmission. Find a discussion of clock gears here: [url]https://www.csparks.com/watchmaking/CycloidalGears/index.jxl[/url]
The involute gear form has come to be the standard for most applications. You mention pinions and gears with small numbers of involute teeth often require the tooth flank to be undercut for clearance to the conjugate gear. The number depends upon the pressure angle but when your tooth count gets to 16 or less and undercut is likely. This reduction in tooth thickness at the root reduces the tooth strength. It is not impossible to drive a conjugate gear with an undercut pinion tooth, it is done frequently. You can make the gear wider to increase the strength.
I cannot think of a good reason that you would ever use a different type of gear geometry on mating gears. So pick one that fits in your package and approximate the tooth loading. Almost all gear tooth stress calculations are based on the "Lewis Equation". This equation gets tweaked based on empirical data by different groups. Your mechanism may have more wear issues than strength concerns. Good luck.