As I argued a few weeks back: the pressure is off. Our players play better and with freedom without that pressure?
That's true of almost all players though. Look at teams who are mathematically relegated ( or all but ) for example and see how they perform differently when the pressure's off. Even the best teams play better when they're 2-0 up with 20 minutes to play.
Mind you, it's a valid criticism which has been made at Moyes' teams and has carried on since and they've been labelled "bottlers". In reality, it's not so much that they've been bottlers, it's more they've been unable to overachieve for two or three games when it really matters ( in cup competitions ).
But Moyes has gone. Martinez came in and, for all his faults, his positivity initially at least, started to change the psyche in the squad. On the surface, Koeman is more pragmatic, but underneath, he's a winner.
Look at the starting eleven today :-
Robles - Martinez man
Coleman - Moyes, but his mentality is spot on
Baines - Moyes, I love him to bits, but probably the weak link mentally
Williams - Koeman signing. Not tarred by the Moyes brush
Funes Mori - Martinez man
McCarthy - Martinez signing
Barry - proven winner with City - Martinez signing
Morgan - Koeman signing. Not been here long but he doesn't seem lacking in self belief
Barkley - Hardly a Moyes man. Brought through very successfully, and it looks like Koeman is toughening him up and hopefully finishing the job. Some doubts mentally though
Lookman - Koeman signing and walked into the first eleven without blinking
Lukaku - Martinez man, I think he'll be fine under pressure
Subs
Mirallas - Moyes signing, but the opposite of bottling in that he tends to show up in the bigger games
Davies - Brought through fully this season by Koeman, but Martinez will have had some influence on him. Has the arrogance of youth on his side which will likely help when the pressure's on.
Holgate - Martinez signing, brought through by Koeman. Unlikely to be affected by the "Plucky Old Everton" attitude from the Moyes era.
Plainly the proof is in the ( pulling of the ) pudding, but mentally, I suspect we've moved on from the Moyes era and both Koeman and Martinez deserve some credit for that.