Currently when vault is locked in the browser and user logs in to new service, there will be no prompt to save new password. As it was discussed in this issue, it is not possible to detect if user already has this service saved in the vault.
I’m using Enpass. I think it solves this issue better, by prompting user to login with pin or master password. Then if user unlocks it, Enpass detects, if credentials provided for this site are new, updated or the same as in the vault. When first two options are detected, user is prompted, if they want to save/update these credentials.
I think it is better solution as it allows user to have vault locked most of the time (e.g. with option “immediately”), but still be prompted.