More detailed information about how LibKey Nomad works and how to set it up is available here
LibKey Link is an extension of PubMed's Outside Tool (recently rebranded as Library LinkOut). If a library uses this service with PubMed, they can configure their link in Outside Tool/Library LinkOut to point to LibKey. The link is shown in a button to the right of the article in PubMed. If your library was already using Outside Tool/Library LinkOut, you were likely already seeing this button and it pointed users to your link resolver.
The button will appear different for each library depending on how they have configured it in PubMed. Other buttons for the journal's publisher or open access sources may also be shown in this same area. These are not related to LibKey and would be something your library has configured separately.
More detailed information about how LibKey Link works and how to set it up is available here.
Because a user can view PubMed with the LibKey Nomad extension installed, it is possible for both LibKey Link and LibKey Nomad to be offering links at the same time. LibKey Nomad is offering the link from the small button near the article's title while LibKey Link is offering the link to the right.