and which name should I use on my Resume when applying on remote jobs, online Identity one or the real ones?
Using anything other than your real name on your resume would be foolish. Your resume is about the real you, not some fictional identity you choose to cloak yourself in. (The famous New Yorker cartoon "On the internet, nobody knows you're a dog" comes to mind.)
In a cover letter or during an interview, you could mention your fake name if that helps identify your work. Hopefully you have chosen a screen name that won't come across as childish or overly eccentric.