It seems like there are two ways to manage who has access to a Sharepoint site and document library, what is the different between them? If I add users to the group Membership they don't show in site/document library permissions. Are there therefore two ways to manage access?
Team Site essentially is an Exchange group + SharePoint site in disguise. Way deeper, it creates a mail-enabled security group in Active Directory that consist of only 2 types: Members and Owners (no visitors). SharePoint then internally in their code, associate the group Members to the Site Members, and group Owners to the Site Owners.
But the traditional Site Owners-Members-Visitors is still there on the site (obviously). Hence, you can manage permission in 2 different ways. But just that it's not very convenient for Owners because it was designed to be easy to manage people in it.
It's recommended to use this interface to add a member/owner. Going through site's members, owners, or visitors just make things more complicated for admin to manage (in my own opinion).