As a site owner, how would I move a list between two sites? The "save as list template" option doesn't appear in the list settings which is where most of the guidance points to.
Proper way does not exist anymore, save as template used to be a life saving option but not available now, you can hack the URL and get to site template page but lists created through template may not work properly.
few options are discussed here
You can migrate any lists or libraries from source(2007,2010,2013,2016,Online) to SharePoint online by migration tools. you can easily migrate them with out loss of any property or data.
here is one of the migration tool. you can download free trail version. this free trail can do your complete list or library with out any problem.Migration tool Download
you have to specify the organization email. Yahoo or gmail won't work for free trail registration.
The general steps are:
Save the source site as a template
Then attach the saved template to the destination site in SharePoint Online
Copy the content from source site to the destination site in SharePoint Online