You can connect your local browser to Till. But, you need to have your browser to accept a custom CA(Certificate Authority). Instructions here: https://till.datahen.com/docs/installation#ca
We don't have a hosted service at the moment. It's all run on premise.
The API link you mentioned is for a Till instance to validate the token and for usage stats reporting for each user. The Auth token validates each user's free and premium features and limits based on usage stats more specifically total requests counts and cache hits counter. We don't record nor track anything related to the requests made on Till.