It all sounds right. Which text is this from? And how is the performance guarantee defined in that text?
If we assume m = 2 in that same proof, why would it not prove that the bound is at least a factor of 3/2 by the same argument, except we replace 4/3 by 3/2. The inequality which reads 2b/m < 4/3 in that proof would then read b < 3/2. And the inequality which reads (2m - b)/m < 4/3 would read (4 - b)/2 < 3/2, that is, b > 1. Since b is an integer, this is a contradiction.
It is the same as the mock argument which I gave above in another post.
I suspect that your recollection is top notch. However, I fear the source of your information cannot necessarily be trusted.
The paper https://arxiv.org/pdf/1807.05554.pdf gives more details on background and more recent bounds. Their new lower bound for the asymptotic competitive ratio is larger than 1.54278.