What is does is basicly the same as you would make a text link. But in addition it allready loads the url allready on the background in a tiny iframe.
The cookie is set the same way as it would when you visit the link. So the plugin does not make the cookie, but the url in the iframe does.
It is not a perfect solution but it does help.
We made it a while ago when we made some ladning pages for some products with affiliate links on it. The affiliate links are where always very basic with a text link: 'more info' or 'order now'.
The problem was that some visitors just keep searching around a little bit on google, maybe for some more reviews or they think they want to visit the manufacturers page. In some cases even understandable.
But when you work hard on a nice landing page and writing reviews it is not always nice when you know you could have earned more on it.
When we implemented this way of linking we did saw an increase of sales generated by the links.
in direct answer to your questions:
Q: are the cookies set by this plugin first level cookies?
A: the cookies, and in some cases just sessions, are set by the url. What the plugin does is basicly allready generating a click on the url.
Q: how does this cookie I set relate to some browser's settings of: block 3rd party cookies?
A: If the browser does block 3rd party cookies or in some cases all cookies. The cookie won't be set. So it's not a perfect solution, but it does help a bit.
Q: so the name parameter is optional or not important.. could be whatever I want?
A: If you want the plugin to generate a link to the url you should set the name. The name is not optional, but you can leave it empty. In that case an empty link will be generated on the background.
If you want to know more, or have any suggestions... please let me know!