Saas requires that the service provider host the customer's data.
TimeAndDate is not SAAS as they don't require data from the customer to provide a service. You subscribe and they grant you access to their API, but it is their data you are accessing.
IMDB is another subscription API service that is not SAAS. You can subscribe to their API and query data about millions of movies, but it is not SAAS because it is not your data.
A valid SAAS example would be bookkeeping. You subscribe, input your accounting data and they host that data on a cloud platform. Updating the bookkeeping software for you, and you pay to have access to your own data. You use their software on your own data.
TimeAndDate charges a fee based upon the quality of the data they provide.
SAAS businesses charge a fee based upon the importance of your data they host. This is why SalesForce makes so much money. There are many businesses that keep their important data on their platform. The more important the data the higher the fee they can charge. It's a way of locking a customer into a SAAS platform.
TimeAndDate can not lock their customers in. A customer can just go use a different API if the data is the same.