Totally agree, even without reading your links. What matters is the actual faith in Gd. I see this myself. We're members of two temples - one more orthodox and one more what you'd call casual.

The people that show up to the casual one once a year or even a few times a year aren't really "different" than someone who doesn't bother to show up. It's good they are there but the religion isn't influencing how they think and act - which is why the TFR is similar between casually religious and casually non-religious.

Where things differentiate is on the extremes. Someone explicitly atheistic (vs just non-religious) has a TFR around 1 from what I remember, while orthodox and ultra-orthodox have 3.3 and 6.6 respectively. What makes the difference is the degree to which they allow the religion to permeate their mundane existence, which is a factor of faith.

This doesn't seem to track with broader sociological trends. For example, let's compare the USA, one of the most religious Western countries, with the USSR, where 60%+ of the population was atheistic, and where the state promoted atheism. Between 1960 and 1980, the US population grew from 179M to 226M (a 26% increase). The USSR population grew from 208M to 262M (a 25% increase). So, despite massive differences in religiosity, the population rate was pretty similar. China, another largely atheistic state, grew from 582M to 1B in roughly the same period - a 73% increase.

So while it may be true that certain small deeply religious populations are more incentivized to have children, this doesn't seem like a significant effect at population levels overall. You'll also find small non-religious groups with similar behaviors.