How to create a circuit to reduce my H-Bridge Inputs from 4 to 2, much like the 4427 Dual Driver does?
You say your H bridge is identical to the one on the link but it doesn't specifically give values for the transistors and diodes - it gives options and ideas so we are missing information about what you actually have. Yes, you can do what you have proposed, the 4 inputs can become 2 inputs. This will give you: -


Off
Forward
Reverse


And if you do some messing with the logic it can give you fast braking where both top (or bottom) transistors are on with the intention of stopping the motor quickly (rather than just letting it free-wheel.

If you are really careful with the design you could use PWM control and get really good energy efficiencies. Especially useful if all you have as a power source is a battery.

Having said all of that, the kudos of building a H-bridge is somewhere between succesfully going to the toilet and shuffling playing cards without dropping any so, why not buy one that does the job. I'm thinking DRV8837 or take a look at TI's portfolio (it includes the 8837) here: -



I've drawn a red block around the most important parameters to consider i.e. operating voltage and motor currents. I meant to add, don't consider either of the following for a low voltage H bridge if you might be taking about an amp: -


L293
L298
SN754410


They just don't cut the mustard these days.