Could someone send me on the right track with a link:www.websitename.com or links:www.websitename.com . I'm trying both and there is a huge difference between the results and I have to admit (dummy) that I don't know what the difference is between the two searches.
I also see a massive difference between the sandbox links: search and regular serps.
The syntax for the special operator is link:example.com - with no space between the operator and the domain. If you use links: or vary the syntax in any other way, you are performing a regular keyword search for the character string.
Your approach is also performing a natural keyword query rather than a special operator query. There is no provision for quotes with the special operator.
So, your results are not showing links, unless the domain name is present as anchor test. Instead those results are showing pages for the search on the full domain name (because of the quotes) and the word "link".