Here's a link that explains back links -
http://en.wikipedia.org/wiki/Backlink -
I think you understand it correctly. Each search engine has a different number of backlinks on record for your site. e.g. Google shows the you have 1 backlink and Yahoo shows that you have 257 (both figures include links to pages on your site from your own site). This is normal, Google usually shows less backlinks than Yahoo. The Google amount of links shown is not very accurate. You can check you links using the commands below -
Google
link:
www.example.com
Yahoo
linkdomain:
www.example.com
In addition, each search engine indexes a different amount of pages from a website and it usually takes time for a search engine to fully index a large website (months and months and months and months and....) - if they ever do.
As VRI mentioned, there isn't a need to submit to search engines as they'll find your site and content from backlinks. Submitting your website to a search engine doesn't mean they'll index your website.
It looks like you currently have 288 pages of your website indexed
http://www.google.com/search?q=site%...L_enUS233US233