Neighbourhood
Contents
Definition
There are 2 common, yet distinct (nonequivalent)^{[Note 1]} definitions of neighbourhood, however what is true for one is usually true of the other too, which is why this hasn't caused a problem (to my knowledge)  both definitions however are common, there is no (obvious) majority.
In both cases we assume that [ilmath](X,\mathcal{J})[/ilmath] is a topological space, and [ilmath]x\in X[/ilmath] is an arbitrary point.
Definition 1
A set, [ilmath]N[/ilmath] is a neighbourhood of [ilmath]x[/ilmath] if^{[1]}^{[2]}:
 [ilmath]\exists\mathcal{O}\in\mathcal{J}[x\in\mathcal{O}\subseteq N][/ilmath]
That is to say:
 [ilmath]N[/ilmath] is a neighbourhood of [ilmath]x[/ilmath] if there is an open set entirely contained in [ilmath]N[/ilmath] where [ilmath]x[/ilmath] is that open set.
On this site we shall only deal with this definition of neighbourhood, if a neighbourhood is mentioned we are using this definition unless otherwise specified  we shall only use the other definition in notes or warnings and so forth.
Definition 2
A set, [ilmath]N[/ilmath] is a neighbourhood of [ilmath]x[/ilmath] if:
 [ilmath]N\in\mathcal{J} [/ilmath] and [ilmath]x\in N[/ilmath]
That is to say:
 [ilmath]N[/ilmath] is a neighbourhood of [ilmath]x[/ilmath] if it is an open set containing [ilmath]x[/ilmath]
TODO: I believe that Munkres uses this definition, but I will check before listing that as a reference
Alec's recommendation
We already have a word for definition 2, it is "open set containing [ilmath]x[/ilmath]", whenever we talk about "[ilmath]x\in N[/ilmath]" using voice, or text, we need only add the word open to convey definition 2.
As all open sets are neighbourhoods to all of their points (as they are an open set [ilmath]\subseteq[/ilmath] themselves) nothing is lost by using the first definition, and we can use the term to describe sets that may not be open, but contain open sets. Which is useful.
As such I come down firmly on the side of definition 1. There is no point to making neighbourhood a synonym to open set.
Author list for each definition
Definition 1  Definition 2 



To confirm:  

Notes
 ↑ This is an example of a Conflicting definition, see Category:Examples of conflicting definitions for examples, that means a view must be taken on which to adopt and use on this site. Read carefully.
References
 ↑ ^{1.0} ^{1.1} Krysztof Maurin  Analysis  Part 1: Elements
 ↑ ^{2.0} ^{2.1} Introduction to topology  Third Edition  Mendelson
 ↑ Functional Analysis  Volume 1: A gentle introduction  Dzung Minh Ha
 ↑ Introduction to Topological Manifolds  John M. Lee