Yes, checking the Elegant Universe book, it was Klein, not Weyl.
But what Weyl did was this http://www.ams.org/notices/200607/fea-marateck.pdf
"In a 1918 article Hermann Weyl tried to combine electromagnetism and gravity by requiring the theory to be invariant under a local scale change of the metric, i.e., gμν → gμν e^α(x), where x is a 4-vector. This attempt was unsuccessful and was criticized by Einstein for being inconsistent with observed physical results. It predicted that a vector parallel transported from point p to q would have a length that was path dependent. Similarly, the time interval between ticks of a clock would also depend on the path on which the clock was transported.
The article did, however, introduce
• the term “gauge invariance”; his term was Eichinvarianz. It refers to invariance under his scale
change. The first use of “gauge invariance” in English3 was in Weyl’s translation4 of his famous
• the geometric interpretation of electromagnetism.
• the beginnings of nonabelian gauge theory. The similarity of Weyl’s theory to nonabelian gauge theory is more striking in his 1929 paper."