Your procedure should not have introduced any air into the system. The reservoir cap is vented so the fluid pushed into the cylinder will just displace air.
With the engine off if you pump the brakes two or three times to deplete the vacuum reservoir the pedal will feel hard. Start up the engine and press the pedal and it will feel spongy because the vacuum boost allows the pedal to go down farther with the same force. This may be what you are experiencing.
The true test is whether the brakes perform as before on the road. It's not clear to me why you folded back the caliper (front) to check the pads. They can be easily checked with the wheel off the car through the caliper window and you can also see the ends of the pads from the front and rear of the caliper.
The important point on the rear brakes is to remove and push back the piston ONE SIDE AT A TIME. Since the rear are fixed calipers wiht pistons on both sides, if you remove both pads and push on a piston the other piston will pop out and can also pop out even if you open the valve.