Here's the equation for rho at the load.

rho = (Z0-Zload)/(Z0+Zload)

SWR = (1+rho)/(1-rho)

I don't see the source impedance in those equations.

Your analysis is fine if the source is matched to the coax, but you are
neglecting the mismatch at the source to coax interface.

If you used a TDR, for example, to look at the set-up you would see 2 points
of discontinuity, firstly at the 100 ohm source to 50 ohm cable interface,
and secondly at the cable to 200 ohm load. BOTH of these discontinuities add
to the overall mismatch as seen by the 100 ohm load.

Your application of the above equations neglects the first discontinuity.

73
Jeff