"publicity" is an important/integral part of scientific research and thereby also mathematical research, but it seems there are not that many scientists and mathematicians who are known to especially epouse it. there are rare exceptions. Carl Sagan comes to mind as a great model/mass popularizer of science who was still held in high esteem within the scientific community. but note that publicity is a concept that also applies within a narrow coterie eg among experts sharing amongst themselves.
in science it is largely handled with the "publication" system [papers], but that is only the more predominant form. conventions, talks, lectures, awards etc (and yes, driving traffic to a site about scientific research via this particular badge you mention) are other forms of publicity. there are some who disparage so-called "promotion" or "advertising" of scientific research/results but that seems backward to me.
an example/case study of a famous mathematician with words/actions that can be interpreted as against publicity: Perelman. but, he's regarded by many within the field as eccentric in this way [among other ways].
speaking of publicity, the general chat room on this site seems somewhat underutilized. maybe the larger community here is not aware of it. my understanding is that the pre-stackexchange version of MO did not have a chat room so maybe this is not widely known. it seems an opportunity for the many elite experts on the site to talk about their research or other aspects of math they would like to popularize. it is easy to post a few brief lines & it supports inline images so graphs or other diagrams can be easily shared. alas it does not support latex but there are installable extensions that support it. would like to see creative uses of it! have posted a few musings already, check 'em out if you have a spare moment.