FWIW, I got that second circuit 'inspiration' from LTC with its easy drive ADCs. See this design note as an example.
And in their App Note, they also say it's for High Impedance sensors.
Delta-sigma ADCs, with their high accuracy and high noise
immunity, are ideal for directly measuring many types
of sensors. Nevertheless, input sampling currents can
overwhelm high source impedances or low-bandwidth,
micropower signal conditioning circuits. The LTC®
2484 family of delta sigma converters solves this problem by
balancing the input currents, thus simplifying or eliminating the need for signal conditioning circuits.
So if you're using their LTC2484 part #, with a high impedance sensor, then you can use their simplified circuit.
Looks like their LTC chip does the balancing.
The bridge configuration is more "generalized."