When we left off, the transition from the back 3 of the W-M formation to a flat back 4 was in process.
One of the inventions that the back four brought along, the double pivot, initially didn't have room for a snarling, hard-tackling, technically limited defensive player, at least not for top teams. In a 4-2-4, the wingers acted as truly forward players, staying high up the pitch and rarely coming back in either the defensive phase or in the building of play. Thus, with the distribution left to the two midfielders, having a technically limited midfield player meant there was too much creative pressure on a single player, a player that could be easily marked out of the game. Of course, a team could distribute from the back, but that meant more long balls, which created its own problems if a team didn't have the necessary players to play long balls or a direct counter-attacking game.
Furthermore, the natural transition from 3-1-2-4 to 4-2-4 saw the defensive half-back drop back to the defence. The ‘2' were the same in the 4-2-4 as they had been in the 3-1-2-4, and so, were initially more attacking as they were the attacking half-back and the creative, deeper inside-forward from the old W-M formation and Brazilian diagonal.
Brazil solved the issue of the 4-2-4's over-burdened double pivot midfield fairly quickly. Martim Francisco was perhaps the first to consciously play 4-2-4 with his Vila Nova side, but, in practise, it rarely appeared as such. This was because it was for too easy for the two in midfield to be overrun, a similar issue to the creative burden a midfield could have. Thus, when in possession, the left-half that had been pushed back to become the fourth defender, stepped forward into midfield to initiate attacks; when the side was out of position, one of wingers, dropped into midfield. Thus, it was never really 4-2-4, but, really, 3-3-4 in possession and 4-3-3 out of position: an amalgam of old and new.
This idea was carried to the national team, winners of three out of four World Cups from 1958 to 1970. Mario Zagallo, a converted inside forward, theoretically played left-wing, but shuttled up and down the flank, giving the team balance. He was happy to become a third central midfielder, so by 1962, Brazil's formation was 4-3-3 even without the ball. Zagallo's positioning meant Brazil could play Zito, a ground-covering but solely ball-winning defensive midfielder, along with Didi, a very creative but not extremely mobile midfielder, and not have a creative burden or lack thrust from midfield; in that way, its composition was not entirely too dissimilar from what would be seen from top-class midfields in the nineties and noughts.
In fact, it'd be easy to stop there, but doing so would ignore the modernizing of the defensive midfielder through two complete opposite styles of football: total football and catenaccio.
For total football to happen, though, pressing had to occur first. And while the development isn't linear, it happened first in Russia, where Viktor Maslov developed zonal marking and perhaps the first true 4-4-2, ahead of Alf Ramsey. Unlike Ramsey, Maslov's 4-4-2 was more proactive, and included pressing; thus, his holding midfield player was very different to the destructive play of Nobby Stiles, and, while the first line of defence to forward players, he was also the initiator of attacks. He had to be the initiator of attacks to take advantage of the space to counter in, and also to allow his side get back into shape. Really, his role wasn't far different from that of a sweeper (libero), who, in most of non English-influenced football, was effectively the holding midfielder.
The sweeper was first deployed by Karl Rappan's verrou system, played by the Swiss team in the 30s. It was a defensive system: Rappan, fearing that Switzerland would be overwhelmed by the technical quality of the central European teams they faced, looked to play an extra defensive player to counteract that threat. Because the best players of teams were often forwards then, the logical place for this extra player was in the defensive line.
That principle was the same for Helenio Herrea's catenaccio Inter Milan: playing a sweeper, and having the whole team drop deep, allowed Inter to soak up pressure, with the libero sweeping up anything that got behind the four defenders. The reason playing so defensively worked for Inter, though, was their ability to counter attack with well-directed long passes. For other teams, though, with lesser players, getting swamped in the midfield was a real problem, and they couldn't counter attack. The solution in Italy, as Lodovico Maradei explains, was that "rather than converting full-backs into liberi, they turned inside-forwards into liberi. This allowed you, when you won possession, to push him up into midfield and effectively have an extra passer in the middle of the park".
The same idea of having the libero push into midfield was one of the key ideas of total football, with the attacking ability of the libero furthered by the vertical swapping of positions and pressing; thus, whoever was playing as sweeper would break up opposition counter attacks if they evaded the first line of pressing. Because Ajax played an extreme pressing game, where everyone defended, and the swapping of positions meant even Johan Cruyff would play in a so-called defensive position, the need for a strictly defensive player was non-existent. Much more important was the ability of every player, even the libero, to play offensive football.
This was somewhat consistent with the move in Italy to convert inside-forwards to liberi. There was an idea that defensive systems that were built to counter attack couldn't work unless the sweeper was a competent footballer; otherwise, the team would get swamped in midfield, and would resort to hitting endless long-balls, and would eventually cave in to wave of attacks, much as Inter did in the 1967 European Cup final against Celtic. And while Ajax were more interested in the technical quality of the sweeper in order to manipulate space, there was again the idea that the offensive thrust provided by the libero was too important; ultimately, as the 60s and 70s progressed, the libero, who was the defensive midfielder of the time, had to be a creative player as well as one who was intelligent in their positioning.