RAW, crusader will get 6 points of healing
Unfortunately, the feat's description is short and simple:
...you make a trip attack...
Rules do include basically all mechanics of a trip combat maneuver into the term "trip attack". They are touch attack, possible attack of opportunity, strength check, and possible counter-trip.
If your DM is strict about the rules, you (assuming success on every list item) then:
- hit for damage;
- make a touch attack and a strength check;
- hit for damage again via Improved Trip.
I think, crusader should get 2 points of healing
We may only guess about RAI in most cases, but there are game elements which are just obviously ill- or under-defined. Knock-down isn't the worst case, but considering slight controversy between its description and benefit, considering my gut feeling about maneuvers' mechanic in the game, I'd say feat actually talks (or should talk) about strength check only, when it says "trip attack".
First, it says in it's description, that it is your mighty blow, what knocks foes off their feet. No need of unarmed attack or an attack with some special weapon, based on the description.
Second, based on similar combat maneuver improver like improved grab:
...without provoking an attack of opportunity. No initial touch attack is required...
...A successful hold does not deal any extra damage...
you shouldn't bother with touch attack, attack of opportunity and counter-trip. You just knock your foe off his feet, or he manages to stay on his feet.
You also shouldn't get more "real" attacks. The thing is that regularily with Improved Trip, you get a "real" attack once - the attack, that actually deals damage. I suppose, an intent here is that just knocking your foe off his feet is too little of a benefit, so authors made it, you are still able to do your melee damage.
It is "trip + damage" with Improved Trip, and with Knock-down it should now be "damage + trip", not "damage +trip +damage". For abovementioned improved grab example it is similarily "damage +hold" and not "damage +hold +damage (as part of a hold)".
If you need additional rules basis for the part about not getting free additional attack, Improved Trip says:
...you [...] get a melee attack [...] as if you hadn’t used your attack for the trip attempt...
so it may be argued, you need to use an attack for the trip attempt to get that second free attack via Improved Trip. Using Knock-down, you hadn’t used your attack for the trip attempt, you had used it for a regular attack (with a rider effect).
Final order of your attack will then look like this:
- hit for damage;
- strength check.