Any link that points to another page could be considered a "backlink" - these doesn't mean every link will show up in "back link reports" from various places.  Every data set is different.

For example: In Google Webmaster Tools, it shows me a handful of links for my site. Bing Webmaster Tools happens to show the same links, plus some that Google didn't. The same can be said about all link tools (Open Site Explorer, Majestic, Ahref's, etc)

You can acquire links through an uncountable number of ways. The type you are currently chasing though may not be the best route for long term strategy or success.  I'd suggest checking out link building related posts here on SEOMoz and then elsewhere.

If you just want to jump into action without really getting a better idea, this site has an awesome list of link building ideas. However, keep in mind there is a certain skill level involved with how you go about building links, something you can't fake easily.

This is another great one. I've built plenty of professional relationships with previous clients (we refer business back and forth), as well as with a few graphic designers who work mostly in the SEM side of things (banner creation, tracking, tweaking to increase conversions).

Guest blogging. It's nothing new, it's nothing that requires a lot of talent, but my god! It does the trick for most clients I manage. Helps me acquires custom anchor text dofollow links from themed sites and the long term relationship with these blog owners are worth their "weight" in gold!  You get a great link, you get some referral traffic, and you expand your network. Triple win!