as I'm not a master in English, I'll try to explain my view a bit differently this time. I repeat, this is only my view and may not be the universal way of categorizing midfielders.
pic #1............................................................................................................pic#2
.......................................
(this is where defensive mfs play.).....................................................................(this is where central mfs play)
example 1.1(makelele/mascherano): plays in the area #1, average at playmaking/attacking, capable of defending.
example 1.2(redondo/alonso/cambiasso): plays in the area #1, capable of playmaking, capable of defending.
example 1.3(pirlo/effenberg/guardiola): plays in the area #1, capable of playmaking, average at defending.
example 2.1(vieira/toure/essien/davids): plays in the area #2, capable of playmaking/attacking, capable of defending.
example 2.2(gattuso, keane): plays in the area #2, average at playmaking/attacking, capable of defending.
1.1 is a DMF, no argument about that.
1.2 is a DMF, I see their playmaking skills as a bonus. (if they would play in a more advanced position, their defensive abilities would be the bonus part of their games that time.)
1.3 is not a DMF, I think we share the same opinion here. not good at defending, not a
defensive mf. it is called regista/deep-lying playmaker.
2.1 is not a DMF, but is a box-to-box midfielder(CMF). they're (more or less) equally good at attacking and defending and they cover much more ground than a DMF does.
2.2 is a DMF, unlike the 2.1 type, because this type of players aren't such balanced players but they are much more focused on the defensive end.
I think you call both of 1.2 and 2.1 type of players as metodisti but depending on where they generally play, I call them either DMF or CMF. I guess, the confusing part was this one.
I hope I made it a bit clearer this time. of course, keep in mind that all of these are my subjective opinions.
