I’ve changed the name of a few things in Hubitat and added a new device (a while ago) and ST will not update (old device names are still there). I did go through the ST app on Hubitat and made sure all the updates were made there. Furthermore, the functionality of a light turning on and off still works, but the corresponding icon (lit up icon for on, etc.) is no longer working. I’m not sure if these 2 things are related.
Follow the prompts to complete the authorization (eg. Login with your Hubitat credentials if prompted, select your location, select your desired devices)
After completing the authorization, I usually suggest giving it a few minutes to complete the synchronization process. Immediately after you successfully complete the authorization with Hubitat, the data will start syncing over and the subscriptions for pushing events starts getting setup.
With a small number of devices, the synchronization happens in a minute, but with a larger number of devices it can take several minutes.
Edit: Also, if you have trouble with the authorization, check out the following troubleshooting guide. Specifically, sometimes a reboot of the Hubitat hub can help. Just be sure to wait several minutes after the reboot to let the hub finish booting and for things to settle down.