When we call the rest API of SharePoint online in SharePoint 2013, we need to perform identity authentication. We could use Microsoft Authentication Library (MSAL).
Official demo:Tutorial: Sign in users and call the Microsoft Graph API from a JavaScript single-page application (SPA)
Similar issue for your reference:
Unauthorized request in Sharepoint API with MSAL token
Access SharePoint Online "_api/web/..." api using the MSAL library
Or you could try to copy the list from online to 2013, and then request this rest API in 2013 directly.
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