"Notification
noun
1. a formal notifying or informing.
2. an act or instance of notifying, making known, or giving notice; notice.
3. a written or printed notice, announcement, or warning:
-----Notifications were mailed to the winners.-----. "
https://www.dictionary.com/browse/notification

I find myself almost always using "notification" where Italian texts (especially legal or bureaucratic ones) use "notifica". This is because it seldom sounds very idiomatic to me to use "notice" at these parts of the text. I presume that the reason for this could be found in this explanation:

"Notice and Notification are two words that are often considered as words that denote the same meaning. ... The word 'notice' is used in the sense of 'observe' or 'warning'. On the other hand, the word 'notification' is used in the sense of 'announcement'. This is the main difference between the two words."
https://www.quora.com/What-is-the-difference-between-notice-...

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Note added at 5 hrs (2023-08-07 14:04:19 GMT)
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Also, note that "service" is being used in your examples as a verb, which acts on the notifica/notification etc. The idea is that "to serve notice" means to serve a kind of warning, while the actual physical letter or form or whatever is provided is a notification.
So, I would say
(a) "to deliver a notification" or
(b) "to serve a notice",
...whereby the first example (a) describes physically transferring the respective letter/form/penalty order etc, while the second example (b) is a more general, yet offiicial, terminology explaining the action in procedural terms.