Hi Evio thank you for posting in the Lucidchart community and for your detailed description of what you're attempting to do and experiencing. This is a great observation - the difference you've demonstrated here is intended behavior. In your first scenario a collaborator accessing the second document via the Document option in the Actions menu needs to have been granted permission to access this document via regular sharing permissions. If they have not also been granted permission to this second document they'll see this error message:

They'll see this even if this second document has also been published because this Document option in the Actions menu requires the user to sign in to access the document within the editor.

The difference in your second scenario is that the Web/Email option in the Actions menu treats the URL to the published version of the second document as it would any other URL and takes you directly to that web page. By its very nature a published document doesn't need to be accessed within the Lucidchart editor which is why users are not required to sign into a Lucidchart account to view it and are able to simply enter the corresponding password. 

I hope this helps clarify! Please let me know if you have any additional questions.