It's even worse. Everyone who you link to has the right to claim damages(in case they are press). So you will have to be careful who you link to since a single link could already fall into the new legislation depending on the member state.
More and more legal entities (governments, companies,...) seem to be moving to actively blurring lines so as to induce fines and damages. Suppose instead the proposal mandated any content hoster (press or not) to include a <linktax> tag so that hosting software can be configured to suppress the inclusion of links to pages that contain the tag.
Without such an explicit tag, the situation will end up similar to ad blockers, with random distributed lists of sources that would probably qualify as "press"
The 'definition' is exactly as vague as I feared it to be:
Consider a blog, one could easily argue that it is a (samizdat) press publication, the blog has a name or title, is regularly updated, provides information to news or other topics and is published as a medium (everything on the web is published in the medium of HTML, PDF, ...) under the initiative, editorial responsibility and control of the "service provider" of whomever maintains the blog.
The same can be argued for any website, forum, government webpage (so we can't even safely link to government pages without exposing oneself to the risk of a fine!)
If the goal was not to expose normal daily citizen activity to the risk of fines, then they should simply require "press publications" to self-declare themselves to be such by means of a <linktax> tag, or perhaps even better, by having those providers who WISH to be freely linked to provide a <linkpermission> tag containing a cryptographic signature endorsing free circulation, such that upload filters can verify without users having to rely on decentralized lists of educated guesses of who would or would not sue for linking freely...