Swapped out a pile of switches in my last house and every single one had the wires stabbed into the back instead of hooked under the screws. Here's what worked for me. First, kill the breaker and check the switch is dead with a tester, because you're going to have fingers and screwdrivers right in that box.
Most push-in terminals have a tiny slot right next to where the wire goes in. Stick a small flathead screwdriver in that slot and push it in firm, that releases the spring clip gripping the copper. While you're holding pressure on the screwdriver, pull the wire straight out. Sometimes it takes a wiggle. If there's no slot, those are the older friction style, so grip the wire close to the switch with pliers and work it in and out while pulling steady. It'll come, just takes patience.
On the question about one wire under the screw and one stabbed in the back, that's just an electrician doing whatever was faster that day. Both make a legal connection on most switches. If it bugs you, once you get the wires free, hook them under the screws on the new switch. Way more reliable long term and you'll thank yourself the next time you have to swap one.