A lot of the information you’re asking about it actually on the page you shared. In case of reaching the limit, the endpoint returns an 429 Rate Limit Error like this:
//Example
{
"status": "error",
"message": "You have reached your daily limit.",
"errorType": "RATE_LIMIT",
"correlationId": "c033cdaa-2c40-4a64-ae48-b4cec88dad24",
"policyName": "DAILY",
"requestId": "3d3e35b7-0dae-4b9f-a6e3-9c230cbcf8dd"
}
Public Apps require a different way of authentication called OAuth. Using this there is only 100 calls in 10 seconds rule beside some other limits regarding the search endpoint for example.
Hope this answers your question.
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Since the daily rate limit does not apply to calls that are made using OAuth, HubSpot doesn’t include the amount of daily calls remaining in the header when for example creating a contact through the API.