lucy24 - 7:28 pm on May 7, 2013 (gmt 0)
The "non-search" version means that the user is logged-in, so there is no query string. It is factually a search, but in both raw logs and piwik it is shown simply as "https://www.google.com" --exactly like a regular referring site. Or, as the case may be, "https://www.bing.com" taking advantage of forums' failure to auto-link https forms.
Again: I'm extrapolating from google to bing. My basic assumption is that google knows how google works. So GA lists all google referers as coming from a search, even when the usual search information is missing. Look at your own raw logs. If you didn't know that google was a search engine, you would assume that someone at a site called google dot com was praising you to the heavens and sending tons of traffic your way.