A payment collected on your HomeRunner website does not update the reservation’s payment status in Hostaway. Nothing about payment flows from HomeRunner back to Hostaway.
Why this happens. HomeRunner creates the reservation in Hostaway with its payment fields empty and attaches your guest’s card to it. After that, HomeRunner does not write to that reservation again. Whatever payment status Hostaway shows on the booking comes from Hostaway’s own charge against that card.
The "paid" status in your WordPress reservations table is written once, when the booking is created, after the card has been accepted. It does not update afterwards based on activity in Hostaway or Stripe. As a result, the two statuses disagree whenever Hostaway has not charged the card yet, or has charged only part of it.
When reconciling. Treat Hostaway or your Stripe dashboard as the source of truth for money collected. Treat the WordPress status as confirmation that a card on file was accepted at the time of booking.