There is a list of Built-in Intents available that can be used in App Actions on the Google Developers website, is there any way we can create our custom Built-in Intents?
No, it's not currently possible to create custom built-in intents for App Actions. As per the App Actions docs:
If these built-in intents do not support your use case, please file a feature request on the public issues tracker
As a side note, while not custom, there is a more generic "open app feature" (actions.intent.OPEN_APP_FEATURE) built-in intent that might work for some use-cases.