want to know whether misdemeanour 4th degree in domestic violence case is the same as disorderly conduct

Never. Never. Never. Although the sentencing maximums for both as Misdemeanor of the fourt degree may be close to the same (fines and days in jail), the term "domestic violence" has dangerous consequences beyond the mere sentencing.  In Ohio, multiple domestic violence convictions can lead to a felony charge the next time someone wants to allege a domestic violence situtation. If the offer is a plea to disorderly conduct, a sane person would take that any day over a "domestic violence" conviction. Good luck.