Combining two columns in Views table
You've structured your data in such a way that this can't be done with a single SQL query. You'll always be asking the database for two separate things: (1) a list of people based on their real names, and (2) a list based on their nick names.

But it doesn't have to be this way. 

The easiest way to accomplish Picture 4, by far, is to always represent a person's "Name" as an additional "Nick Name".  

So "Daina" should have 4 Nick Names: "Kristi", "Rita", "Sweetie", and "Daina". Then, all you need to ask the database for is a list of nick names.

This probably isn't the answer you're looking for. But I would argue that it's a more correct way to store your data. Think of the "Nick Name" field not as a list of alternate names, but rather as a list of all names that the person is known as. Based on the requirements of your situation, that's what it sounds like you're really after, anyway. It's a more complete model for your name data.