I am assuming that at the Webhook setup screen, you set up the URL of the API provided by your server as "Webhook Endpoint"
but that did not work.
Later you use Zapier for "Webhook Endpoint" instead and the only add records event works fine,
so I think you set up Webhook correctly.
What I am concerned is you selected other events, such as Record is edited/deleted/Comment is posted, at Webhook setup and saved the setting,
but maybe you have not updated the app -clicking "Update App" at the end.
Also, you might want to check the "Webhook Execution Log" about the situation. If there is no log, you may have forgotten to update your app settings.
•Setting up a Webhook
•Webhook Execution Log
I hope it helps.