The Sanskrit word Sannyasin names a person who has taken Sannyas Diksha from a qualified sampradaya guru. Details vary by sect. Two things hold: the Sannyas Mantra is received, and worldly aims are renounced.
A Sannyasin of the Adinath Sampradaya could not marry, take gainful employment, rent or own property, or keep dependents, and wore the cloth of a sadhu. Shri Gurudev Mahendranath took that diksha in 1953. He was the last guru of that seat. He would not give it in the West.
The International Nath Order does not confer Sannyas initiation. Its initiation is a householder transmission. A Nath or Devi may work, marry, and keep a home. Sadhu names the renunciate life. Sannyasin names the person who has taken that particular diksha. Neither word is a synonym for Nath in this Order.