Webhook data is based on the underlying API, in this case the Get Streams endpoint. Because of this the type field, along with all other fields, will always be what the API would return. Having the field include the type of change that triggered a webhook notification would only have use within the context of webhooks, to anyone just querying the API having a field that says game_change would make little sense.
It may be possible for notification trigger reason to be added outside of the body data, perhaps in the headers just as the Twitch notification ID and timestamp are there, but that depends on what is allowed within the webhook specification, and is an idea for the webhooks team to think about.