In my book there are questions where there are two intersecting sets $A$ and $B$ and the questions begin by saying something like "In the given Venn diagram ... ". But there is no Universal set surrounding the two sets $A$ and $B$. Just the two sets themselves. Strictly speaking is this kind of diagram an Euler diagram or a Venn diagram ? Does a Venn diagram need a Universal set ?
The distinction between an Euler diagram and a Venn diagram is not the presence or absence of a universal set, but in the relations that are represented. In a Venn diagram, every possible intersection is shown, even if one or more of those intersections is empty. In an Euler diagram, only "interesting" (i.e. nonempty) intersections are shown. For example (blatantly stolen from Wikipedia):
A Venn diagram:
An Euler diagram showing the same information: