Yes, we're all confused following the bizarre overtime coin toss sequence that resulted in the Patriots kicking-off in overtime.
Because of the confusing dialogue between Matt Slater and referee Clete Blakeman, most initially believed that the Patriots asked to kick the ball off when they actually intended to receive it:
This assumption was incorrect. Here's what we think happened:
Bill Belichick wanted the defense to take the field first and indicated to Matthew Slater to kick in a chosen direction. The Patriots chose heads and won the toss. Matthew Slater was supposed to "defer" the decision to the Jets. Instead, Slater asked the referee to kick off in the chosen direction and not defer. Because of this, Blakeman allowed the Jets to choose the direction, which is what confused Slater (when he says, "don't we get to choose", and points towards one end zone).
Bill Belichick's response was typical, saying he "thought it was the best thing to do."
Slater himself confirmed that the Patriots' intention was to kick-off - saying he checked with Belichick a few times.
So yes, Slater misspoke when he asked to kick the ball off, but not nearly as badly as was the initial thought. The bigger gaffe from the Patriots was the overall decision to kick instead of receive the football.
Considering the Patriots were without their two starting safeties, and were very close to giving up several big plays throughout the game as a result, this was not a smart call. Yes, the Patriots were terrible on offense and asking the unit to go 80 yards with a makeshift offensive line would be a tall task. However, considering the wind and conditions were not a big factor, this was a coaching mistake.
The Jets, as you would imagine, went down and scored a touchdown on the opening drive of overtime, thereby denying the Patriots a possession.