In U.S. practice, the terms are used as follows:
Litigator - an attorney whose practice is focused on litigation, that is lawsuits. This is mostly used to refer to people who specialize in civil suits (suits brought by someone other than a criminal prosecution agency) seeking an award of money or an injunction (a judicial order) requiring someone to do something or stop doing something.
She was a litigator specializing in intellectual property trial work.
Counsel - This is a general term for lawyer, also known as counselor. It is frequently used to refer to a lawyer who works exclusively for one company or one agency.
He was general counsel to the NBA.
After the outburst, the Judge said "Counselor, control your client."
Attorney - this is a general term for a lawyer in both the federal and state court systems in the U.S. In general, attorneys in the U.S. are permitted all types of practice (with a few minor exception) unlike the solicitor/barrister systems elsewhere.
The attorneys who graduated from that law school went on to various jobs as litigators, criminal lawyers, corporate lawers, and in-house counsel.