Which image shows up when a link is "attached" on Linkedin?

Add these meta to your webpage head tag.

<meta property='og:title' content='title'/>
<meta property='og:image' content='image/path'/>
<meta property='og:description' content='description'/>
<meta property='og:url' content='your url'/>

While posting on the Linkedin, there is a linkedin cache so if you are going to re-post the same url again, probably the meta will not be picked up. To fix that you can add a parameter at the end of the url while posting on linkedin

e.g original url www.example.com then you can repost this with www.example.com?1

The LinkedIn share functionality supports a small subset of open graph tags as documented here: https://developer.linkedin.com/documents/setting-display-tags-shares

Note that at least until recently (and possibly still) re-scraping of existing shared pages is broken, so if you've already shared a page and it got the wrong image you're stuck unless you share it with a different URL.

LinkedIn posts can also be designed from scratch using the jssdk. I created a tool that allows me to select what image is displayed, and what caption gets set, when I share a link.

If you want to develop your own, fork my repo:

https://github.com/kaburkett/LinkedIn-Advanced-Share