While I tend to err on the side of tradition when cooking as antiquated of a staple of human life as rice, I give you credit for recognizing the flavoring potential inherent to its steaming process - it's often overlooked in my experience.
While there are no rules in cooking as far as I know, besides maybe respectfully representing your ingredients, you'll find that the traditions for flavoring steamed rice all adhere to universal principles of maximizing flavor, texture and last (and probably least) plate appeal.
My all-time favorite rice dish with certain Chinese, South-East Asian and Caribbean meat dishes is coconut rice. If you were to make this by adding liquid coconut flavorings to cooked rice, you would not only alter the texture of the rice, but these flavors would not be given the opportunity or time to permeate the rice.
This is similar to the age-old, "yesterday's tastes better" story - herb, meat and vegetable flavors seep into the broth of soup or stew, spices further aromatize rice pudding and so on.
The quintessential example of a rice dish that requires flavors to permeate rice is risotto. I haven't had day old risotto but I can guarantee you that the flavor will be all there and then some. (It's not such a great reheater I don't think.)
And besides my mom, from Kuwait, where rice is a fairly diurnal form of sustenance, says never add liquid (besides maybe some oil or an acid) to rice after it's cooked or it'll become soggy, so there! (I feel like Adam Sandler's character in that college football movie he did.)
When I make coconut rice, I steam the rice with coconut milk and water, and fold in coconut butter and fresh grated toasted coconut just before serving to embolden the coconut flavor and add a textural as well as visual counter-point respectively. If I added these ingredients before cooking this wouldn't be possible.
When my Mom makes biryani, she adds slivered almonds, raisins and fried onions but as a dressing for cooked rice.
All cultures, including my Mom's, also understand the importance of accentuating flavors in ingredients and this is why you see (or hear) rice dishes that start with a sizzle at the bottom of a rice pot, where a flavor base is built.
In the case of Chinese fried rice, this occurs after the steaming because the flavoring elements are not just there to flavor the rice, but are components of the dish unto themselves, that look better, taste better and have better palate feel when flash fried, rather than steamed.
Many Arabic, Indian and Caribbean rice dishes steam rice with whole spices, such as cardamom, bay leaf and cinnamon to infuse the rice with flavor not dissimilar to the way you infuse water with flavor when you make tea. That said, they are not then served with a dish that only contains these flavors, so to not have the two dishes clash. I agree with Spork that you'd be better off serving naked white rice than using teriyaki flavoring in it, if that is also what you're serving on your birds.
While a steamer is handy when you are serving a crowd, I adhere to the tradition of cooking my rice over the stove so I can have every opportunity to build a flavor base or steam it with liquids other than water. If I'm in a hurry I break with tradition with a pressure cooker.
