Is it required to publish a Chrome extension to the Chrome web store in order to use Google's OAuth?

I'm trying to use the chrome.identity.launchWebAuthFlow method to authenticate the Google user's login in my chrome extension to get his email address, and my code is:

Manifest.json

"key": "Key_here",
"oauth2": {
    "client_id": "my_client_id.apps.googleusercontent.com",
    "scopes": [
        "https://www.googleapis.com/auth/userinfo.email"
    ]
}

popup.js

function startOAuth() {
    chrome.runtime.sendMessage({"message": "start_oauth"});
}

$("#login").on("click", startOAuth);

background.js

var manifest = chrome.runtime.getManifest();
var clientId = encodeURIComponent(manifest.oauth2.client_id);
var scopes = encodeURIComponent(manifest.oauth2.scopes.join(' '));
var redirect_uri = chrome.identity.getRedirectURL();

chrome.runtime.onMessage.addListener(
        function (request, sender, sendResponse) {
            if (request.message === "start_oauth") {
                chrome.identity.launchWebAuthFlow({
                    "url": "https://accounts.google.com/o/oauth2/v2/auth?" +
                            $.param({
                                "scope": scopes,
                                "response_type": "token",
                                "client_id": clientId,
                                "redirect_uri": redirect_uri,
                                "prompt": "consent"
                            }),
                    "interactive": true
                }, function (response) {
                    console.log('response: ' + response);
                });
            }
        }
);

But when I click the popup icon, I get in the background's console:

response: undefined

Unchecked runtime.lastError while running identity.launchWebAuthFlow: Authorization page could not be loaded.

When I open the url in the browser:

https://accounts.google.com/o/oauth2/v2/auth?scope=https%3A%2F%2Fwww.googleapis.com%2Fauth%2Fuserinfo.email&response_type=token&client_id=<my_client_id>&redirect_uri=https%3A%2F%2F<extension-id>.chromiumapp.org%2F&prompt=consent

I get the error:

Error: redirect_uri_mismatch

I'm loading the extension as unpacked extension, have put the extension's id, which appears under its name in the chrome://extensions/, in the Application ID in the Google API's credentials

But under this field it said:

Application ID

Last part of your app's Chrome Web Store URL

Does this means that I have to publish the extension to the Chrome web store in order to make the redirect_uri match ?

Note:

To get the extension's key, which I'm putting in the manifest and referring to it as "Key_here", I uploaded the manifest of the extension to the chrome web store developer's dashboard, as a draft, and got the public key from there and put in the manifest.

Now the redirect_uri is the same in the the Google API's console and in my unpacked extension id, what am I missing?