Comparing Notifications: Pop-ups vs. History

There appear to be two distinct components of Windows notifications: the brief pop-ups that appear for a short period of time (referred to as "toasts") and the notification history accessed by clicking the Notifications icon. Is it possible to manage these two elements separately? For instance, I find the pop-ups from Macrium Reflect notifying me of backup progress useful, but I do not require a log of these pop-ups. Conversely, in a scenario where I was configuring task scheduler to execute a script before the computer goes into sleep mode, a pop-up notification would be irrelevant, but having a record of the notification could offer valuable diagnostic insights. (In this particular case, either the script or the notification failed, as evidenced by the lack of information in the notification history.)

Is there a method to independently control these distinct facets of notifications? Currently, the only option I can identify in the Settings is a binary choice for enabling or disabling specific notifications.