Single player:
OP, sorry, but you are basically completely wrong. If you have a shorter 1h weapon on Horseback you are not meant to charge another enemy with a polearm. If they are mounted, you maneuver behind or to their side until you can close, and then maintain close combat with them engaged in a duel. You will win. I do this all the time. I'm not big on lances.
If they switch to a bow, make sure you ALWAYS close on their right side.
If you have shield, you are sure for the win.
Trick is making sure you either have the most maneuverable or fastest horse. If fastest, choose an approach angle and gauge their course. Close in. If most maneuverable, close in and maneuver so you are in their rear left or right quadrant use a steady gate that allows you superior turning. Use side stepping and quick break or acceleration.
If they are on the ground with a pole-arm, you'd best dismount. Then you are superior, unless outnumbered.
Mulitplayer:
You're not completely wrong.