After researching it a bit, I thought it might be of interest to post my findings here. First of all, so that we know what we are talking about, this is the Clergy tartan that most people are used to seeing.

There is a green version, as well, that you will sometimes see. But of the different versions of the Clergy tartan on record, these are the only two commonly woven.
The Clergy tartan is often said to be the only tartan intended for use by an occupation, and not a clan, family, district, corporation, etc. There is long standing and oft-repeated statement that the Highland clergy wore Highland garb, but were told to eschew the brighter colors. Now, I've never been able to determine if this was indeed true, but have no reason to doubt it. James Scarlett, who knows as much about tartan as anyone living, is of the opinion that the Highland clergy wore whatever tartan the local hand weavers were producing, just like anyone else.
Anyway, when we start getting into the nineteenth century, with industrial weaving and set named patterns and the like, we find the first record of a tartan named "Priest." Wilsons of Bannockburn, the primary producer of tartan cloth at the time, and the first mill we have record of that named their tartan patterns, has a record of a tartan by this name -- our best guess is that they simply thought "Priest" a suitable name for a muted tartan. Theirs was a tartan that was black, lavendar, and light blue. (No. 246 in the International Tartan Index, or ITI).
James Logan apparantly got hold of the design, and reproduced it in his The Scottish Gael, published in 1831, under the name "Clergy." But his design was a bit different. He seems to have gotten one of the pivots wrong, and changed light blue and lavendar to white and grey. (No. 1823).
William and Andrew Smith published The Authenticated Tartans of the Clans and Families of Scotland in 1850, and included the Clergy tartan, as well. They attempted to reproduce the tartan as given by Logan, but with the original lavendar and blue colors. They used a color that would pass for lavendar, but they sometimes mistakenly used it for two or three stripes that should have been black. Also, the light blue in some copies of their book turned out a greenish grey. Variations occurred from one edition to the next, and sometimes between copies of the same edition, so this caused a lot of confusion later on. (Nos. 39 and 111 are both from the Smiths' work).
The next time the tartan was illustrated was by James Grant in 1886, in The Tartans of the Clans and Septs of Scotland. He illustrated the light blue lines right, but the lavendar in the original had turned into a clear blue, and he used it for two lines that should have been black, apparantly copying the error from one of the Smiths' books. But in the text for his book, he says that the Clergy tartan was white, black and grey. This seems to indicate that he intended to illustrate the tartan from Logan's work, but the publisher substituted a different illustration. In later editions of his book, the text described the tartan as dark blue, light blue, and black, but in the illustration this time light blue was rendered as green! This is probably where tartan 701 fits in.
Lastly, in the first edition of The Setts of the Scottish Tartans D. C. Stewart attempted to make a compromise between the original Wilson sett and Logan's sett, but this had the undesired effect of creating yet another variation on the market, so in later editions this was amended.
The connection with the Clark tartan is due to the fact that both the surname "Clark" and the word "Clergy" have the same root in the Latin word clericus from which we get "cleric" and "clerk." The Clergy tartan seems to have been used by the Clark family for that reason. In fact, in many of the nineteenth century works I cited, the tartan is called both names. The practice today of producing the Clark family tartan in different shades than the Clergy tartan is probably just to allow for distinction between someone wearing the tartan because they belong to the Clark family, and someone wearing the tartan because they are a minister.
So then, who can wear the Clergy tartan? Well, as I have said before, there is no such thing as an "entitlement" or a "right" to wear a tartan. There are no laws where this is concerned. But just because you can wear any tartan, does not mean that you should wear any tartan. In today's culture, tartan is generally understood to be representative. When you wear a tartan, you are identifying yourself with what that tartan represents. Why you may choose to wear a particular tartan is completely up to you -- but you don't need to be able to present a pedigree to prove your "right" to do so.
In the case of the Clergy tartan, wearing this will imply to people that you are involved in ministry. Out of respect for those who actually are ordained clergy, most people would consider it very innapropriate for a non-minister to wear this tartan.
But for those in the ministry, any Clergy tartan will do. Just wear the one you like the best (though you will find that if you want anything other than the blue Clergy tartan, you may have to have the cloth woven.
I have heard it said that certain variations of the Clergy tartan are for Catholics and others are for Protestants. This is unfounded. To my knowledge, the Clergy tartan has never been restricted for members of one particular sect or denomination. Of course the two main religious bodies in Scotland are the Presbyterians (Church of Scotland), and Catholics, followed third by Anglicans (Church of England).
The only denomination-specific tartan that I know of is the Episcopal Clergy tartan, designed (according to the notes in the ITI) "by Rev. John B. Pahls, 1966, to honor the clergy of the Scottish Episcopal church and of the Episcopal Church in the United States of America and to commemorate the bicentenary of the death of the Right Reverend Samuel Seabury, first American bishop."
So, if you are Episcopal Clergy, you might want to wear that tartan, or the other Clergy tartans. But other than that, the Clergy tartan can be worn by any man of the cloth! Not that members of the clergy have to wear Clergy tartan. I know many ministers and priests who wear their clan tartans. I have often thought that a solid black kilt would look stunning with clerical dress. And I have sold one "Dark Douglas" kilt (Lochcarron's black on black version of House of Edgar's "Dark Isle" tartan) to an Anglican minister, who wanted a solid black kilt, but also wanted a tartan.