How did these battle happen? In SP?

In SP everything can happen.

If you have not experienced these things more than a couple of times you need not to worry.
Trust me, if you do 5 fights with a unit of Lancers vs Kats. where you are the Kats and then 5 the other way, you will notice the Lancers will win every time. If not then something extremely lcuky happened, such as the Kats killing the other general with the charge.

The point is that Lancers and Kats have the same charge bonus, but the Lancers have a better attack, thus their charge is stronger.
Their stats are: Kats (8/3/5/7/4) (charge/attack/defence/armour/morale)
Lancers (8/5/7/9/8)

So in all aspects the Lancers outdo the Kats. There must have been another factor playing in there.

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BTW, Danish Crusades are true to history.

You may not care about war, but war cares about you!