Cavalry was mainly for scouting, raiding, reconnaissance etc but time to time they did fight with other cavalry, or were used to push the retreating enemy so the infantry could catch up. They tended to fight dismounted in an open skirmish order formation. Charges, as you say were tough but they did occur numerous times but usually with bad results for mounts and men, unless it was Cav on Cav. They had sabers but preferred numerous pistols, and usually carried a carbine or short shot gun as well. Sabers were not the preferred method of fighting unless you just had to.

Now, Mounted Infantry is just that. They rode horses, and carried their long infantry rifles. When it came to a fight, they dismounted and fought with tactics and formations just like the infantry. Usually they were in a cavalry brigade or force and the cavalry would then push the advantage of the Mtd Infantry or come in and screen while they made their withdrawal. As with cavalry, they were utilized more for scouts, recons and raids, but did have the ability to step up and fight pitched battles with enemy infantry units.

Hope that is clear as mud, but they were still infantry, they just didn't have to slog around like true "leg" infantry. Kinda like modern day mechanized infantry that rides around in armored vehicles and dismount as required vs. the light infantry that tends to walk everywhere.