I disagree that it's undemocratic to vote for Brexit without specifics. There are problems with the concept of referendums in general but IMO the Leave vote was largely motivated by racism. Running another referendum is basically saying "here's another chance to not be racist." Brexit is a royal screwup but I doubt a lot of the people in Rural Britain™ that voted Leave were looking at trade accounts.
If the referendum was taken in good faith as an advisory then I agree that it need not be specific to be "democratic". A 52:48 result interpreted with cross party parliamentary support for the softest of Brexits would be democratically consistent. The problem was that Theresa May tried to find a Brexit that satisfied her divided party but not parliament. Parliament was willing to support Brexit as they showed with executing article 50. The rhetorical narrative that anyone not supporting a Hard or No Deal Brexit are not respecting the democratic mandate of the referendum is simply that - rhetoric. An unspecific advisory referendum is entirely consistent with a parliamentary democracy and if Parliament can't find a compromise then General Election is the reasonable way forward under status quo conditions, not default disruptive exit. Any other course is undemocratic by UK constitutional convention.