The Combat chapter of the PHB (p. 195-196) has this to say about shoving a creature:
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
The target of your shove must be no more than one size larger than you, and it must be within your reach. You make a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you win the contest, you either knock the target prone or push it 5 feet away from you.
Here, the word "shove" is defined as either pushing or knocking prone.
The Charger feat (PHB, p. 165) states:
When you use your action to Dash, you can use a bonus action to [...] shove a creature.
If you move at least 10 feet in a straight line immediately before taking this bonus action, you [...] push the target up to 10 feet away from you (if you chose to shove and you succeed).
Ignoring the "moving 10 feet" part, the bonus shove granted by the Charger feat seems likely to give the option to knock prone or push, as normal. The interesting part is when you do move at least 10 feet before the shove and succeed, as the text says that the target is pushed, but shoving is still, as its core, a choice between pushing and knocking prone.
If a Charger Dashes as his action, moves the 10 feet towards the target, then shoves as his bonus action but chooses to knock the target prone instead of pushing, would the target also be pushed up to 10 feet away?
It would make sense from a thematic standpoint (the charger ramming the enemy, sending them flying away and landing on their behind!), but from a mechanical standpoint, I'm not quite sure.