I think the usage directly relates to the history of the movement.
"Jehovah's witnesses" (With a lower case 'w') started out more as a descriptive term than the name of an organized religion. JW's actually would deny that they were a religion if asked. They claimed every faithful Bible character from Abel onwards as their own. (i.e. They were all Jehovah's witnesses) A good example of this occurs in the 1959 publication, Jehovah's Witnesses In The Divine Purpose in the chapter, " A People With A Most Ancient History"
During this period, "One of Jehovah's witnesses" was the only correct usage.
Somewhere in the 1970's this changed somewhat. I can't put my finger on the exact moment, because there was never anything resembling a formal announcement. At any rate, the spelling changed to "Jehovah's Witnesses" (With an uppercase 'W') and the term bacame the formal name of a religion.
As with most subtle changes of this nature, I suspect the real motivation was legal; perhaps attaining recognized status, owning property, etc.
{Edit} == Just realized this duplicates some of Leolaia's comments