If you are going to tell that story, then you should tell it truthfully.
When Reagan was negotiating with the democrats (who had both houses) he struck a deal with them that he would agree to raise taxes only if they would cut spending 3 to 1. The democrats agreed. However the democrats never did go thru with the spending cuts they promised. Reagen regretted that deal with them.
The very same thing happen to Bush Sr in his "No New Taxes" pledge. He was duped by the same trick. Taxes increased, spending cuts never happened.
BTW that's yet another myth pushed by the liberal MSM.