Technically speaking, logging in and logging off only affect an identity cookie stored within your web browser.
Which brings up a bit of a misnomer in the back-side statistics. "Distinct Logins" and etc is actually the number of distinct identity cookies the web server recorded that day. Which is why the number continued to climb at a pace which exceeded other statistics: often, people create an account, log in, get the identity cookie, post a message, and then go back into lurk mode for weeks or months. Other than demonstrating this effect, "Distinct Logins" was pretty useless. As most of the back-side statistics proved to be.
The difference between front-side and back-side. Front-side is the first Free Republic program your web browser contacts. The front-side valiantly defends the back-side from becoming swamped by attempting to satisfy your request by consulting it's massive cache of documents. However, should the requested document not reside within the front-side's cache, the front-side can do nothing else but to pass your request to the back-side (from which it will often remember the page in order to satisfy subsequent requests.)
If you're still following, it should now be obvious that the back-side didn't see enough of Free Republic traffic to accurately report statistics. And hence, the discontinuance of the back-side statistics.