Air carrier First Officer reported descending below charted altitude on approach to SAT; citing lack of situational awareness as contributing.

Narrative

Upon reaching 'QUESO;' at 4000 feet and 210 knots for the ILS13R SAT; ATC cleared for the approach. From QUESO to NCLDA to NUGIT it is a 90-degree turn to left. The FO/PF asked the CA/PM to query ATC if they needed any specific speeds; which they did not. The FO/PF reached the FCU and selected 190 knots to further reduce speed in preparation for configuring. A few moments later; the CA/PM said 'we are cleared for the approach;' reached the FCU; selected 3200 feet (altitude in NUGIT); pulled for 'Open Descend' and armed the approach. With that; the FO/PF thought that the FO/PF might have missed something justifying the CA/PM intervention on the FCU. The CA/PM seemed concerned about being high on the ILS; and towards NUGIT both pilots agreed to set 2200 feet (FAF altitude for the GS intercept) on the FCU. With that; both NCLDA and NUGIT were crossed around 300-500 below the prescribed altitudes (4000 feet and 3200 feet; respectively). The FO/PF realized the unintended CA/PM error when the CA/PM said We are a little low." At that point it was just outside the ALAMO (FAF); which was crossed fully configured for landing. The approach inside the FAF was stable with normal landing. ATC never queried about the altitudes nor did it give "a number to call."The CA/PM and FO/PF talked about it; and the FO/PF showing the approach chart; asked the CA/PM what was the thought process behind selecting 3200 feet and open descend. The CA/PM thought for some reason that NUGIT was the next fix after QUESO; and got concerned about being high. Both pilots discussed about the temporary break in Manual;and ultimately Situational Awareness (SA); due to break in communication."