Pushing on the bar in the direction you want to turn is one way to initiate countersteering, which is the primary way you steer/lean the bike.
What you do with your own weight from there is your own business, but there are some strategies you can use. Keeping your body in line with the bike and letting the steering do all the work of establishing lines and lean angles is probably the least tiring approach in most situations. Any other approach comes down to whether you want to accomplish something specific with the lean angle of the bike, because what matters is the bike-and-rider overall CG, and if your CG moves one way, the bike's can move the other to result in the same path at the same speed, provided you steer accordingly.
A lot of people like to put body weight on the outside of the turn in low-speed maneuvers so they can lean the bike more for a given maneuver.
Going into a corner too hot, your first response should be to countersteer more, which will tighten the turn and increase your lean angle. It is possible to get to the point where you drag parts, and if you're even close to that point, you don't want to be leaning out of the turn. Leaning into the turn beyond the angle of the bike allows the bike to straighten up, gives more clearance, and allows a tighter turn before you drag parts (Traction limits, of course, still apply).
I've also been known to lean slightly into a crosswind to put the bike more upright and let the suspension operate on a more vertical axis.