Not quite.
If that person attempts to legally marry another while currently legally married that is bigamy - an illegal act.
Simply fornicating or cohabitation with another is not bigamy. The key is attempting to legally marry more than one person at a time.
You are changing the hypothetical situation I was responding to. That situation had "people living together, considering themselves as married, as in a group marriage or polygamous marriage." And I stand by my statement of the law that would apply in that situation: If one of the persons in that group is legally married and lives together with the other persons in it as husband and wife, that married person is engaged in polygamy. The reason is that in most states, at least, the cohabitation as husband and wife between the person who is legally married and other persons to whom he is not legally married does not create relationships to which no law applies. Instead, it creates common law marriages to those persons.
Bigamy and polygamy are two different things. Terms commonly used interchangeably in layman discussions, but two distinct things under the law since persons can be in only one Civil Marriage at a time, but can (it varies by state) be in additional non-civil marriages at the same time.
State laws make the two separate crimes, although the Supreme Court has blurred the distinction (see below).
In many states one can be legally married to one spouse and cohabitate with that spouse and another and as long as there is no attempt to legally marry both at the same time it legal.
I question that. As I said, if a legally married person lives together with more than one other person as husband and wife, it constitutes polygamy under most state laws. If you know of states where it would not be polygamy as long as the married person lived together as husband and wife with only one other person, what are those states?
Neither bigamy nor polygamy has ever been favored in this country. In an 1878 decision, Reynolds v. United States, the Supreme Court considered the case of a Mormon in Utah Territory who had been convicted under a federal law for purposely marrying a second woman while he was already married. The law said that "every person having a husband or wife living, who marries another, whether married or single, in a Territory, or other place over which the United States have exclusive jurisdiction, is guilty of bigamy." The Court upheld the law, rejecting the man's claim that it violated his right to free exercise of religion.
It is interesting that the man plainly considered what he had done to be polygamy, but never tried to deny that the federal bigamy law applied to it. The Court noted that at trial he "proved that he had received permission from the recognized authorities in said church to enter into polygamous marriage." And the Court referred throughout the decision to what the man had done as "polygamy":
"Polygamy has always been odious among the northern and western nations of Europe, and, until the establishment of the Mormon Church, was almost exclusively a feature of the life of Asiatic and of African people. At common law, the second marriage was always void, and from the earliest history of England polygamy has been treated as an offence against society....
From that day to this we think it may safely be said there never has been a time in any State of the Union when polygamy has not been an offence against society, cognizable by the civil courts and punishable with more or less severity."
Polygamy has been so strongly opposed in this country that Congress required several territories, as a condition of being admitted to the Union as states, to pledge in their state constitutions to ban polygamy by law forever.