Will marriage without divorce be called live in relationship?

Hello,

In your case wife is not in live in relationship. She can be punished for second marriage under ipc. I will suggest both the parties to mareiage to file mutual consent divorce. 

This is a case of bigamy. The husband needs to move the Court, esp. because  of the birth of the child,

You have to prove the fact of her second marriage. Then only she gets punishment under Sec. 494 of IPC. If you can't prove the fact of marriage you can prove fact of Adultery by taking copy of Birth certificate of her child provided she didn't use your name as child's father.  But nine months before the birth of child Law of Adultery should have not been repealed. 

As Law of Adultery is now repealed you please ensure that it was not repealed as on date on which the accused person probably had intercourse with her paramour. 

If you use birth certificate of child it will be easy to prove the offence of Adultery under Section 497 of IPC. 

Considering the fact that "Wife had married another man", her relationship won't be called live-in relationship. 

Section 17 of the Hindu Marriage Act, 1955 clarifies that bigamy will be punished according to Sections 494 and 495 of the Indian Penal Code, 1860.

Section 494 of the IPC provides that bigamy will be punished with imprisonment upto 7 years and fine.

If your wife concealed from her second husband that she was already married at the time of 2nd marriage, then she will be punished (under Section 495, IPC) with imprisonment upto 10 years and fine.