Thank you, Emm, for the legwork.
From:West's Encyclopedia of American Law
The common law also established the "marital paternity presumption," which holds that a child born during a marriage is the offspring of the husband. Therefore, a child born as a result of the wife's adulterous affair is recognized as a legitimate child of the marriage. This rule recognized that illegitimacy brought social stigma as well as severe economic penalties to a child, including the inability to inherit from the husband of the child's mother. By establishing a presumption of paternity and therefore legitimacy, the rule promoted family stability and integrity.
This is was in my mind after reading some of london's comments, but I would not have bothered to do all the googling. This is the argument pro marital paternity presumption that I have heard before, too.
My understanding is (sorry, no sources to support my statement) that in some countries the possibility to recognize someone else than the husband of the mother as legally the father of a child is a rather recent development. That is, if the marriage is still valid and the married couple has no intentions of divorcing. Thus those legislations accept that the social stigma of being an illegit child has lessened in the recent years and sometimes it is best for the child to have their biological father acknowledged legally, even if the mother is married to someone else. This requires consent and agreement of all adults involved, as I described in my previous post.
However, after all this discussion and the quotes provided by Emm, I'd say that most likely the OP stands pretty strong if she wants her husband to be recognized as the legal parent of her child. The BF has less to say than the husband, in most cases. I'd still strongly recommend the OP to check the local laws and get the knowledge of what to expect.