Cavall is right. Rules-wise it would be two separate charge actions, each requiring fulfillment of their requirements to use.
Interpretation-wise, you are right to assume that the charging character need not have stopped running or charging, just as the turn-based round-by-round movement isn't meant to portray a character walking across a room as stopping and starting every 6 seconds in a herky-jerky motion or even moving in a grid-based 5-foot step pattern. That's just the game system, but this is the Rules forum.
It wouldn't be unreasonable for a GM to allow such continuing momentum in certain cases (feat-based and whatnot). There would have to be rules to cover some situation, for instance, there's no rule that you have to attack forward in a charge, though that's what's typically envisioned (and usually what happens since you move to the closest square) but there's no change to the bonus or anything if your target somehow ended up in a square beside or behind you when you made your attack, even though the bonus is clearly generated by your momentum towards the target. So if you did this, you would have to make rules against allowing a second charge attack against the previous target you rode past (but might still be in range for a reach weapon) and then move further away. But that's all just interpretation and homebrew.
