No, it's not authenic, but it's close. Of course, if you ask 50 Pilipinos how to make any traditional dish, you'll get at least 45 different answers.
For your recipe, add 2 Tbs of cooking oil to the list of ingredients, and use it to sauté the chicken pieces, onion, and garlic until chicken is brown on both sides before you add the liquids and cook it down.
I would also add 1 tsp crushed red pepper, cut the quantity of soy sauce in half, increase the vinegar to 1 cup, and combine it with 1/2 cup water, but that's just me.
Oh, and Pilipinos never serve their food over rice. That's the Chinese. Pilipinos put the rice on the side, and then flavor it with the sauce from the Adobo. Which reminds me, you can also add big chunks of pork butt to the pot to make it a combination chicken and pork adobo.