Short version:

Is it possible to detect that someone added code to run inside a page from the browser inspector?

Long version:

Stock broker companies give their users the real time value of stocks, other free tools give you a delayed version of such values, for example 15 minutes old information.

There are other types of financial companies that have real time API to give you access to stock market at a cost.

What some people do is to keep their browsers open in the broker site and inject some JS code to observe the changes and post them elsewhere using XHR or web sockets. Not only network calls but also notification API and the draft Serial API can be exploited to put data out of the site.

This usually can't be done automatically due to the secure nature of logins requiring captcha or other methods. But once logged in and injected the hack will work until the tab is closed.

Usually this is not done by injecting script tags with outer files source, just pasting the whole code inside inspector and running it.

Now back to the question: Can a site know that code rogue code is running in their site?

I thought of some methods like a HASH of every variable used and if anything new is created it reloads the page or warn the user. But I'm not sure it is possible in nowadays JS, I guess document.all could help.