Update: Op followed the advice, below, and changed the question from a bug to a feature-request.
This is not really a bug. Right at the top of the API documentation, it states that the API is read-only, in bold letters:
Stack Exchange API v2.0
This is the documentation for the v2.0 read-only (with optional authentication) Stack Exchange API. If you have additional questions, or believe you have encountered a bug, don't hesitate to post a question on Stack Apps.
And neither the /me/inbox, nor the /notifications/unread docs claim to alter the state of the inbox items.
And, yes, that note seems to be misleading since the API does allow modifying comments. But, the docs do not claim any write ability for notification state.
For now, if you wish to avoid rereading "new" inbox items, you must track them in your app.
This might be better as a feature-request, rather than a "bug", since no claimed feature is not working.
If you absolutely want a bot to change the read state, you'll have to use standard page-scraping and/or userscripting techniques to automate the appropriate clicks -- until such time as this inbox-reading feature is added.