No it isn’t.
A dirty bomb is merely a conventional bomb used to disperse radioactive debris. Lots of things are radioactive. Some soil is radioactive. Gather up a bunch of radioactive garbage and pack it around a large conventional bomb and you have a dirty bomb. It works better if the radioactive debris is dangerous enough to cause panic.
A nuclear bomb is an energy release derived from a nuclear reaction. Energy is created out of matter. There may or may not be deadly radioactive debris produced as a byproduct.
I understand that.
At the same time, I understand that U.S. military policy is not to respond to a conventional attack with WMD. And that, for this purpose, WMD included "dirty bombs".