No, it says "Offenses against the United States" (Art. II, sec. 2), which means federal crimes only.
Your wording is correct. I went by memory. Wikipedia and other online sources state that it means, as you say, the President’s power to pardon is for federal crimes and the Governors can pardon State crimes. I have heard Constitutional experts say the fact that there has so far not been an example of a President giving a pardon for a State offense the issue has not been tested. As a matter of fact, with the exception of impeachment the power is universally accepted as “almost limitless”.
Your point of view limits the power of POTUS to deal with only Federal law while others and I say the it is the Governers who are limited. They can pardon only State but not Federal crimes but the pardoning power of the President is limited only by the one exception of impeachment.