Are the elements of a module also called vectors? Are the elements of a module also called vectors? Or if someone says 'vector', are they talking only about a vector space? If no context is given, are there some standard assumptions?
 A: No.  Normally we do not have special names like this ...
Elements of a group are called groupies
Elements of a ring are called ringlets
A: I wouldn't call this an answer, it is just my opinion.
The elements of a given vector space I would only call vectors, if the vector space is $k^n$.(Probably I am missing something...) Why? For example I would not call matrices, polynomials or functions whose codomain is a vector space vectors.
On the other hand given a generic vector space I would call the elements vectors. For instance I apply a norm to vector.
I am tempted to say I call vectors $v \in V$ a vector if $V= \bigoplus k$. Now there is alway an isomorphism so for me it is legit to call elements of a generic vector space vectors, but given a specific vector space it should be "canonically" $\bigoplus k$.
Now if I transfer this to modules over a ring $R$, I think it is fine to call elements of modules of the form $\bigoplus R$ vectors, but in general I would not call the elements of a module vector.
