Ok mate. I'll keep saying a tactic which top coaches say they use is a valid and you keep saying it's not valid. Wonder which is right. The one supported by the likes of Diego Simeone and Jurgen Klopp or yours.
As usual, it's pretty much impossible to have reasoned discussion on this forum without it descending into a farce.
I think you are conflating some different tactical ideas.
Also using the word "concede" in terms of possession is also a bit misleading.
Counter attacking football is a tactic, of course and for sure any coach can speak to it, and some very well known coaches have set-up teams to play the style either all the time or in some games. But counter attacking is also a simple expediency that occurs naturally in the game, for example when possession is conceded to the defending team at a corner you often see the team in possession counter attacking to take advantage of the open space and the oppositions players being out of position.
Your suggestion that teams allow opponents to maintain possession so they can counter attack is true. A good example would be Mourinho's Chelsea when playing Man United at Stamford Bridge in spring of 2015. Van Gaal was always going to try to maintain a high level of possession but we had seen over and over again that United struggled to turn that into a real goal threat. Mourinho let United have all the ball they wanted, and focused on breaking up their play in the final third, then launching fast counter attacks. Chelsea won 2-0.
Liverpool under Klopp do not do that, neither did he do that with Dortmund. Simeone does not do that. Pressing the ball is not the same as counter attacking. If Klopp has said he wants to allow opponents to have possession in certain positions he is certainly speaking about pressing defenders into playing the ball into a bad position which then allows a press that can easily result in a turnover in possession in a dangerous part of the field. Fir example balls played to full backs where our players can cut off lines and press the FB. Often that results in an FB either just hoofing it forward or trying to dribble out and losing the ball. That isn't counter attack.
KK its good that you are interested in tactics and style of play. You should read up more on it. Look for interviews with top managers, you can learn a lot.