There seems to be several problems.
You need to use
<script src="https://apis.google.com/js/client.js?onload=googleApiClientReady"></script>
instead of
<script src="https://apis.google.com/js/client.js?onload=onClientLoad" type="text/javascript"></script>
because you need to make sure the initialization function googleApiClientReady() in auth.js is called.
However, the Google API also reports that there exists no OAuth client with the ID trusty-sentinel-92304.
Edit:
If don't have an OAuth client ID, but rather an API key, you shouldn't use the auth API at all. Instead, you need to add the API key as parameter to each API call (as described here: https://developers.google.com/youtube/v3/docs/standard_parameters).
Try this as a start:
gapi.client.load('youtube', 'v3', function() {
var request = gapi.client.youtube.search.list({
key: "YOUR_API_KEY",
q: "cats",
part: 'snippet'
});
request.execute(function(response) {
var str = JSON.stringify(response.result);
console.log(str);
});
});