For the most part, that's true.
However, there are unauthorized uses of the API markings ...
API has found motor oils in the marketplace (shown below) that are displaying the API engine oil certification marks without API's authorization. API has not reviewed a licensing application for these oils, and the oils have not been approved by API as meeting any API engine oil standard and are...
www.api.org
I looked for Traveller oil and couldn't find it on the API site, however the API search tool is very sensitive to spelling inputs, corporate names, etc ...
I generally believe that Traveller oils are licensed as they claim.