This is more work to audit than I have time to handle. The difficulty is mostly that any random wiki could have a gadget / extension which adds a message on arbitrary criteria... and even for just the core / extension code, there's enough different ways to trigger something like this that it's a real pain to search for it happening.

That said, here's an analysis of what VE adds and on what criteria:

VisualEditor itself will show (this all applies to the VisualEditor-source-mode (NWE) as well):

• WelcomeDialog (the switching dialog), with modes for if it's switchable or not
  • Doesn't show on protected pages (i.e. requires wgIsProbablyEditable || wgRelevantPageIsProbablyEditable)
  • Depends on the wgVisualEditorConfig.showBetaWelcome preference which defaults to true
  • If the user is logged in, the user-option visualeditor-hidebetawelcome will suppress it, which is set after the user sees the dialog the first time
  • If the user isn't logged in, local-storage and cookies are checked for ve-beta-welcome-dialog which suppresses it, and is set after the user sees the dialog the first time
  • Also suppressed by the vehidebetadialog URL parameter
  • Desktop only, even if mobile-VE is being used
• Notices (the /!\ toolbar icon)
  • Local notices - explicitly added by VE
    • If wgTranslatePageTranslation === source, warn about being on a translatable page, which VE doesn't support
    • If on a not-explicitly-supported browser, warn that there might be problems
  • Remote notices, sent from the server
    • Suggesting anonymous editors log in
    • If the URL parameter editintro was present, show that page's contents as a notice
    • Warning if there's a newer revision of the page
    • Warning if the wiki is read-only
    • Notify if you'll be creating a new article
      • Warning if the new-article title is protected
    • Warning if the article is protected from editing
    • Warning if *you* can't edit for some reason (IP block, etc)
    • ...and whatever else would be shown above the classic editor. There's a hook for this, so any extension can add some.
• Education popups (the pulsing blue dots)
  • VE itself only applies this to the link tool
  • The Citoid extension adds it to the cite tool
  • These are shown/hidden as a group, not individually, after any education popup has been viewed and closed:
    • User logged out: ve-hideusered local-storage/cookie is set
    • User logged in: visualeditor-hideusered user-preference is set

If the classic WikiEditor is being used, and VisualEditor is installed, those same WelcomeDialog display checks are applied.

Mobile has an entire different editor implementation, and so misses most of these. If logged out, it'll show the "we'd like it if you logged in / created an account" dialog.