Make a decision: What do we want to do when we lose the ball in the final third? What is our priority? To press high and regain the ball quickly or to simply close down the first attacker and gain the few precious seconds that will allow us to regain our defensive shape? I'd prefer the former.
A stop here for two quick explanations before anyone jumps on me: We do press in the sense that we want to incorporate active pressing tactics. It's clearly an arrow that Solskjaer wants us to have in our quiver. And we are slowly but steadily becoming better at it but the caveat here is that, currently, it bears fruit when we're already in defensive shape and the opposition decides to build attacks from the back. We got two goals from this type of pressing tactics yesterday. Secondly, as it has been proven in the big games, our defensive transitions can work to a tee when the instructions are to quickly drop off and get back in our third of the pitch.
Our main issues occur in the games in which we want to commit more bodies forward to stretch defences and we also want to stay higher up the pitch to apply more pressure and, eventually, break down an opponent that can defend well. What i took from Solskjaer's quotes in the OP is that the area we need to improve is our immediate reaction when we try something that doesn't pay off and the opponent gets on the ball. From what he suggests, we can argue that he wants us to stay high up the pitch and try to regain possession. City, Liverpool and Spurs under Pochettino are pretty good at it and it has been a key element to their successful seasons. The moment they lose the ball, they begin their effort to get it back. And it's something that their frontlines really excel at.
We don't get that from our attacking/creative players much, at least not all of them. The end of an unsuccessful run in behind or dribble/cross/shoot sees most of our players throwing their arms up in the air or with their heads down in disappointment. Pressing doesn't work like that, it's a collective effort and if one player doesn't play his role in it, it means that an opposition player will be free on the ball and we'll be finding ourselves running like crazy to get back into shape. Notice on how many occasions Fred or McT rush to close down the first attacker but are swiftly being overrun because their teammates simply don't operate in the same wavelength as them.
This involves two phases of play: Firstly, when we overload the left side with Shaw, Fred, the CAM and Rahford to get the latter closer to the box so that he can link-up with Martial. Lots of our attacks break down there and, as it can happen with overloads, your weak side is susceptible to counter attacks. In our case, with the #10 out of position (to be involved in the attacking play) and with Martial staring at the ceiling (not always and he's improved lately but let's not pretend that it doesn't happen often), what we get is one of James/Greenwood not knowing who to close down first, AWB worrying about getting back into position, McT caught between two minds (should i press or should i drop deep?) and Lindelof trying to anticipate what will happen in front of him. This also happens a lot when we're static in the attacking half and one of our midfielders/defenders decides to play a through ball or make something happen in general. Again, these are not bad players (i mean bad-bad like some want to portray them). They're not the best in the world (and most of them could be upgraded, if only for our dear Ed) but they could do a decent enough job. When the instructions are clear (drop deep into a low block and we'll hit them on the counter), they're not easy to get bypassed.
The idea of a specialized defensive midfielder who will screen the back-four suggests an inverted triangle in the middle of the park and from what i get Solskjaer loves his 4231 with a designated #10 who will enjoy the cover of a double pivot behind him to influence the attacking plays by his movement between the lines and by linking up with the main forward (which includes runs in-behind since the forward is instructed to create too). It's one of his principles and i don't think he will give up on it easily. Then you'll ask why Carrick could do it for Fergie? Well, then we had decided to defend in a very low block and with a very narrow shape ourselves and there was enough creativity and firepower in the first team to get us going on the other end of the pitch. Now, we simply don't.
A part of the blame could be laid at the feet of our centre-halves too. Not fast enough, not very aggressive and not very urgent. The thing is, they were selected by Solkjaer for a purpose. Their ability to help in the build-up phase when the midfield (especially with Pogba as a CAM) consists of ball-winners and runners (and the rumoured transfer targets are players of the same ilk). So, what can be done when you have slow defenders and no designated DM and you want to keep a high line? The answer is: press better and apply pressing that leads the opposition to play the ball to the sides. Don't allow the direct long-ball down the centre (the one the DM is supposed to defend against) and don't allow through balls in behind your slow defenders.
I believe this is what Solskjaer was referring to, the need to become even better at pressing. Thus the quotes about (hopefully) improving during this season and with these players. Imho, he should insist. If we get the likes of Martial, Rashford and Pogba to buy into this, the team as a whole will suddenly look much better. I'm one of those who believe that he has made far more mistakes than he should be allowed in the implementation of his tactics but his general idea to move away from Mourinho's "get all men behind the ball and keep the shape at all costs" primary tactics and instil some intensity and aggression in the attempt to get the ball back instead is a step towards the right direction.
The problem is that, as usual, we have made things difficult for ourselves. We want to maintain a higher line with defenders that are relatively slow but our main attacking outlets (Martial, Rashford and Pogba) are not very good at regaining possession right after the ball is lost. And we have a midfield that can react when we lose the ball adequately by either pressing high or dropping deep but they're not exactly sure what they're supposed to do in each occasion.
* Note regarding yesterday's game against Newcastle. In their goal, we actually led them to the wide areas and although we didn't get the ball back, we managed to get more than enough defenders behind the ball. But nobody closed down Longstaff, they all stood there watching. That's on the players.