The key is that because the 'Leave' campaign weren't in government and weren't one single organised group and didn't actually hold power they could promise literally anything safe in the knowledge that they'd never have to actually deliver it. In fact after 'Leave' won the referendum the Prime Minister resigned and none of the key Leave campaigners actually mounted a serious campaign to replace him- someone who campaigned for remain was put in charge almost unchallenged.
So through all this murkiness you end up with one key fact: 48% of the country wanted to remain, 52% wanted some form of leaving but each of those options I listed above probably garner no more than 20% of the vote by themselves. So we're 52:48 in favour of leaving, and 80:20 against any specific way of leaving.