How can I wait for an ajax request and process the result?

You can wait on both simultaneously and handle whichever occurs first:

await Promise.race([
  page.waitForNavigation({ waitUntil: "networkidle0" }),
  page.waitForSelector(".Error")
]);

if (await page.$(".Error")) {
  // there was an error
} else {
  // the page changed
}

The accepted answer isn't waiting for a request but waiting for a selector. That kind of hacky-way wasn't doing it for me.

Waiting a promise resolution coupled with the request response status is the easiest way to achieve it.

await Promise.all( [

  page.click( selector ), // ... action

  page.waitForResponse( response => response.status() === 200 ), // ... wait for network

] );

• Source @ https://github.com/puppeteer/puppeteer/blob/main/docs/api.md#pagewaitforresponseurlorpredicate-options

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HTTP response status codes

• Source @ https://developer.mozilla.org/en-US/docs/Web/HTTP/Status/200