In the last 10 years, defensive midfielder is probably the position that has the most evolved. It can be justified by different facts but its definition is still changing, and what is expected from such a player is now far different from what it was a few years ago.
First of all, we can state that there is various kinds of defensive midfielder, and that defending is not their only mission. The perfect example of this extended role is the box-to-box player, broadly used in English football, made popular by players like Patrick Vieira, who dedicated himself to get the ball back from opponents and to go forward as soon as he could. Born in England in the last century, this box to box role has broadened to international football during the nineties. French football has widely integrated this position in its forming centers, and provoked the advent of powerful and endurant players able to attack like Paul Pogba or Maxime Gonalons.
Another kind of #6 is able to take part in the offensive game while evolving in front of the defense, it is the organizing defensive midfielder, or faux numero 10, symbolized by the wonderful Andreas Pirlo. Contrary to the role upper mentioned, this kind of players have a more strategic than defensive role. The positioning is essential for them, in order to find good passing solutions and allow them to cut offensive passing lanes. In EPL, Arteta has been doing miracles for years under the Everton outfit and now for the Gunners in this role. Yohan Cabaye is also evolving in this register, even if his defensive role is preponderant.
The opposite of the faux numero 10 is probably the old school #6, which tends to disappear, and was symbolized during its last years of reign by Gattuso. Nowadays, some players perpetuate the tradition of spending the whole game running after the opposite number 10 and contesting every ball aggressively like currently does Mathieu Flamini with Arsenal. Even if this type of player is highly valuable for a team, is it not enough when you are a defensive midfielder to focus on the defensive tasks because you are also expected to bring some vista and fine passing to the all-defensive package. That is why this kind of player is generally associated with another defensive midfielder more focused on play-making, as does Arsene Wenger by making play Flamini with Arteta (faux numero 10) or Ramsey (box to box).
The requirements of modern football have forced players to add offensive responsibilities to their defensive duties. Like Blaise Matuidi, whom his natural vocation is to protect his defense but is more and more dedicated to take a part in the construction of the game. The versatility of the midfielders is a growing phenomenon in modern football, it is now common to see a defensive midfielder take part in offensive actions. There are probably different reasons to that. The main one is the advent of the Spanish and Barcelonian model, which gives to the midfielders a multidimensional role, the essence of the total football : everybody attacks and everybody defends. The borders between defensive and offensive midfielders became blurred and the logic was focused on what the game needed you to do, giving players much more responsibilities. That is also why everybody cannot play in this scheme, because it requires a lot of discipline. The other reason to the evolution of the role of defensive midfielder in certainly the takeover of the English Premier League. Not of English football, but of its Premier League which is the most followed football league worldwide and composed of the best players. As I said before, the EPL promoted internationally the box to box players few years ago, while the rest of Europe played with a #6 and a #10, with totally different responsibilities.
The time where a footballer played only in one position is now done. More than ever, versatility is a requirement to be considered as a valuable player on the market (except for some positions like pure goal scorers). With the example of the #6 position, we can see that the players make the game, but the game pushes permanently the players and the market to evolve.