From the American author @Denise Dianaty whose novel "The dance plays on" is one of my favorites.
Yes, "the Cloth" means the priesthood, usually of the Catholic church. It is a very old colloquial turn of phrase; it's pretty much only used when writing in historical settings as in my book. Back then, there weren't so many Christian denominations. So, using "the Cloth" could still be broadly understood as the career of being a priest.
No one says it that way today.
The modern vernacular refers to them as priest, pastor, minister, or preacher – depending on the particular denomination's or community's predilections.
It's partly because there are so many denominations, and partly colloquial, and partly due to the particular duties. All priests don't preach, but a "preacher's" job is to deliver sermons from the pulput… i.e., "to preach;" very often, preachers are not full time and have other careers simultaneously. "Pastor" derives from the concept that the local church leader watches over and cares for his/her church community – they provide "pastoral service" to the parish or congregation and community; this includes things like visiting hospitals, charity outreach, home visits, funeral services, etc. "Priest" is the oldest term, I believe; it is a more generic term covering all the other titles, but can also be monastic and is always male only in Catholicism and orthodox churches. "Minister" is roughly interchangeable with "pastor;" the minister provides ministerial care to his/her congregation; s/he ministers.
Again, the choice between "pastor" or "minister" or "preacher" may also be a local colloquial use. But, the terms may also be used interchangeably. My pastor may be referred to as "preacher" by someone else in the same church. I might refer to him/her as "minister" at times. It's kind of fast and loose how the terms are used.