Which leave campaign? There was a few of them and they didn't give a consistent message. None actually have any power to implement anything so how can they make pledges anyway?
Also, that link has the picture of the bus as the first image... a 'pledge' completely based on lies that was dropped literally minutes after the vote.
See I'm not a brexiter/leaver/whatever (Personally I think it's a terrible idea), but it's blindingly obvious to me that a lot of leave voters (maybe even the vast majority) thought the main issue was immigration and that Mrs May had a choice either exit the single market, or basically ignore the result of the referendum, she chose the former and not the latter.
UK spent 40 year entangling with the EU, disentangling cleanly in 2 years is a pipe dream.
It is still a possibility though; declaring for hard brexit could very well be a negotiating tactic (as in 'nothing left to lose') as May hinted to the possibility of a transitional agreement.
The challenge for me is that soft-brexit is basically no brexit 'cause EEA access == sign-up to the four fundamental freedoms and accept the primacy of EU law, which seems to cover most of the things that the leave campaign wanted to get rid of.
As you say this could well be a negotiating tactic though, and no-one will really know until they start the negotiations. I think it's entirely possible that once the details are fleshed out people might seek to change it.