Segments get a bit interesting when you're trying to exclude stuff. There are two ways of doing this - using "does not contain/equal" or creating an "excludes" container.
What you're doing right now is the first one, using does not contain. Stuff like this works better at a hit level than at a visit level. Basically, this segment is saying "show me a visit where there is a hit that does not contain the link name". So if someone has two hits in their visit, one with the link name and one without the link name, because there is one hit that matches the criteria it includes it. But there is also a hit that doesn't meet the criteria. I created a sample segment to demonstrate. For mine I just used a basic "page doesn't equal home". But when I put that in my table against visits and break it down by page, there are still home page views appearing. Basically, the way segments work is, if it meets the conditions specified, it returns all the information at that level. This segment is saying there is a page that doesn't equal home. Then you return the entire visit. And if they say a home page and a not-home page, it will return all the pages in the visit, including the home ones.
What you want to do instead is use an "exclude". Change "does not contain" to "contains", and then click the settings gear at the right hand side and select "exclude". You should see a red line on the left side of that sub-container. This basically says, if the condition is met, exclude all of this information. You want it to be at a visit level. So "exclude visit where link tracking contains link name". And you can see in my sample table, there are now no home page visits at all.
