Lots of different ways to counter someone as a pressure fighter. A lot more than just being a backfoot counter striker. You can force the action and dictate the pace a lot more.
Countering while they are throwing:
1) Be in range and play the jab game. Exchange jabs and make them think that's what you are doing, then you time their jab, slip to the inside and come over the top with an overhand simultaniously, for example. That's the cross counter, but don't be too predictable with it.
Illustrates a few variations of that counter.
2) Slipping to the outside of their jab instead and jabbing with your own
Using the space:
1) If you are fighting/sparring someone who wants to get away from you while you are cutting off the ring and pressuring them, leading them into either side then catching them with either (shuffle step) lead hook, or a rear hook, depending on which way they are moving is an option.
Drawing out punches:
1) Volume, feints and intet. People being pushed back and overwhelmed sometimes opens up a lot just because of the pressure. So be ready to counter and strike. Also feinting, and putting your head where they think they can hit you, while deliberately waiting for the attack to slip/block/parry and counter, is a staple. The pull counter is an excellent one for that (hard to do though), or slipping to the outside and coming back with the rear uppercut or straight afterwards. You have to read the opponent and that can be hard at times, it's a chess match.
2) If you can pressure them enough to go highguard, and have them against the ropes, then going to the body is your best friend. Just be careful of their counter hooks.
Changing up your tempo, look and level:
1) Be unpredictable. Don't do the same draw each time. Step back sometimes and give them space, take angles, don't come straight in. Use headmovemt. Go high, and low. The more you confuse them and give them different looks, the harder it is for them to read what you're doing and when you are letting them punch, or countering. Sometimes it's okay to give them a false sense of security, lull them a bit and then bang.
Keep your eyes on the opponent at all times. Being an aggressive counter puncher is all about being aware, using your eyes and reading the opponent.