contrary to places like HK where the ambassy and the HK governments have agreements and work hand in hand...
Here what you do with the PH ambassy only has relevance with the PH, not with SG.
You don't need to do anything like contract authentification to be in good order with SG MOM...
You'll need the formalities with the PH ambassy when your maid wants to go home for holidays and only then. And then when you do that you'll discover for example that you can't change anything on the standard contract they want you to sign even if your maid earns more than the professed 600 SGD...