Full Idea
Nowadays Russell's position is routinely put by saying that existence is what is expressed by the existential quantifier and only by that.
Gist of Idea
Existence is entirely expressed by the existential quantifier
Source
report of Bertrand Russell (On Denoting [1905]) by Colin McGinn - Logical Properties Ch.2
Book Reference
McGinn,Colin: 'Logical Properties' [OUP 2003], p.20
A Reaction
We must keep separate how you express existence, and what it is. Quantifiers seem only to be a style of expressing existence; they don't offer any insight into what existence actually is, or what we mean by 'exist'. McGinn dislikes quantifiers.