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.