Captain reported the flight crew received a domestic CPDLC departure climb via clearance and mis-read the cleared altitude on the CPDLC format; resulting in an altitude deviation. ATC confirmed the cleared altitude on the SID with the flight crew; then re-cleared the flight to a higher altitude.

Narrative

We [had] a CPDLC (Controller Pilot Data-Link Communications) Pre-Departure Clearance; CLEARED TO TEB ARPT BLZZR5 THEN AS FILED; CLIMB VIA SID; EXPECT 13000 ft." We both misread the clearances and understood it to be except 13;000 ft. We set the altitude preselector to 13;000 [ft]. When contacting Departure; I checked in with present altitude of 2300 [ft.] climbing to 13;000 [ft]. We received a TA with traffic that was above us. We thought it was due to climb rate; so we reduced the climb to avoid any further alerts. The Controller then asked what altitude we were climbing to and I repeated 13;000 ft. She then said we were given 5000 ft. and to continue climb to 13;000 [ft].Suggest adding; "TOP ALTITUDE XXXXX;" to the; "CLIMB VIA SID; TOP ALTITUDE 5000; THEN EXPECT;" or eliminate; "EXPECT ALTITUDE;" altogether; or change to; "REQUESTED ALTITUDE 10 MINS AFTER DEPARTURE." CPDLC Communications should be able to detect incorrect altitude in the ALT PRESELECTOR. Also; Controller should have caught that I was climbing to wrong altitude on the check in. We had just made a flight from ZZZZ. On their SID's; it requires you to check in with; "CALLSIGN; SID; CURRENT ALTITUDE; CLEARED ALTITUDE." Maybe we can add this domestically. Thanks for helping improve the system and eliminate opportunities for errors."