Neither of those amount to a purely counterattack style. In 20/21 we scored much more goals from established play than we did from counterattack, and we generally had more possession than the opponent. Even against the top teams it was usually close. And compared to our main competitors we fared relatively better against lower table (and generally low blick) teams.
You are right I guess - depends on how you define a purely counter attacking style. I'd say, we didn't really have any intended style, it was just that Ole was a pragmatic coach, trying to achieve the most with basic tools. Defending is the easiest when you do it as a deep block where your players are close to each other. Attacking is the easiest when you have room to make use of. So When you combine that, you defend deep and attack fast against teams that failed to penetrate you deep defense. That is what counterattacking is - and this was what Ole did to some success. But as time went by, opponents noticed that their chances for points would be low when they tried to attack us. So many decided to defend deep themselves against us - it was a viable plan because a) we weren't particularly good defending against counters (not many teams are), b) we were especially tame when in possession and not able to create chances and c) smaller teams were happy to take a point from a draw. (you can see this development in a microcosm in the games against City, we won when Pep underestimated us or our capabilities in terms of counter attacks, but in 2021 he didn't give us room in all the games that really mattered and shut us down with it).
This development was what broke Ole as he wasn't able to create a style of play where we could make use of the possession other teams surrendered to us. And while I am aware of the infamous 2nd place in that season, many indicators where there that we weren't doing particularly good on the field (visible i.e. in xG, xGA) even in comparism to teams that ended slightly below us in the table.
To finish the circle: you are right, we haven't had a deeply engrained counter-attacking style of play. But we didn't have anything else at all. The players today aren't failing (until now) ETH because they are more suited to counter-attacking football. Those players are just somewhat limited technically and in terms of being organized on the pitch. And this is where ETH comes in hopefully.