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Hi Akiko,
Sure, it's possible to sync Security group from on-premises AD to Azure AD so the Group type shows Security. Also if you update members on premises, it also will sync to Azure AD. It's an expected behavior. But for Dynamic attribute-based Distribution Group in Azure AD, it will filter group members automatically based on the rule you created, which is different from Security group.
For your reference: https://docs.microsoft.com/en-us/azure/active-directory/connect/active-directory-aadconnectsync-understanding-users-and-contacts
Of course, based on your description, the Security group synced from on-premises AD can meet your request. We are glad to hear that and if you have any concern about it, you are welcomed to come back to us.
Thanks,
Iry