e.g. if I trigger an update locally, from CI or from somewhere else, I should get a Slack notification. Is this possible?

That is possible if use the Pulumi Cloud. We have Webhooks support that allows you to trigger whatever you want. Here is a hook handler sending a Slack notification. https://github.com/pulumi/examples/tree/master/aws-ts-pulumi-webhooks

From my understanding, the example pulumi stack creates an API gateway route that forwards incoming events to Slack, but how does Pulumi forward the event data to that API gateway endpoint?

If you go to

<https://app.pulumi.com/><your-organization>/webhooks/new

, you should see a form like in the screenshot. The payload URL should be your custom endpoint receiving the webhook data. In the example case, this is the endpoint of the API gateway you deployed.