In the Markl, Schneider and Stasheff text, topological operads are an indexed collection of spaces $O(n)$ for $n \in \{1,2,3,\cdots\}$ satisfying some axioms. In May's text, the index set is allowed to include zero. 


1) Is there a standard terminology for operads with and without $O(0)$? 


2) Is there standard terminology for topological operads where $O(0)$ is a point, vs. $O(0)$ not being a point? 


Although it's less important I'd be curious if people have examples where these distinctions are interesting. 


Since any operad acts on its $O(0)$ part perhaps the $O(0)$ part should be called something like its "base"? But then "baseless operad" would sound kind of pejorative. 



Operad terminology - Operads with and without O(0). 
I can second Jeffrey's comment, reduced is used to say that O(1) is just the monoidal unit (it allows us to use the Boardman Vogt resolution in homotopy theory). It's my opinion that this terminology will probably stick.


I would also say that a $\mu$ in O(n) had arity n.


That the O(0) part of an operad is referred to as the 'constants' of the operad makes a lot of sense, every algebra for O must contain O(0) and the composition of those must behave in a certain way.


Calling O(0) the point also makes sense, because in the category of algebras O(0) will be the initial object.


*Here my comment has become too long, just as I've got to the point of my comment:*


The comments to the question tend to prefer terminology that relates to the behaviour of the operad (eg "reduction", because a unit lowers the arity). My personal preference (and I think the literature follows it), is that terminology should have more of a relation to the category of algebras than to the operad itself.


So my vote is that you call O(0) the initial of O. And you call an operad without O(0) initial-less or uninitiated.


