but if I delete all the emails from the server, why is there a list of all the subscriptions I've received at that address?
It's a historical list created as emails arrive, not a list of subscriptions currently in your mailbox. If you unsubscribe from the newsletter's website (so Microsoft doesn't know you unsubbed), it may eventually drop off the list but will take a long time (months).
Your example indicates that the email client or webmail would need a button to indicate something is a newsletter.
It's in the message header. The sending server marks it as a subscription (as well as bulk mail). This is something all bulk senders are supposed to do per CANSPAM laws.
As for the ones you say are not newsletters, check the message headers to see if there are properties identifying it as a newsletter or bulk mail.