#1 John who took my pen is here.
#2 John who is wearing my jacket is here.
How firm a grasp would you like to obtain? (I know I shouldn't have written that, but I couldn't resist.)
The thing that works for me is to take out the the subject and the "verb" and see if it still makes sense. (That may even be the "rule.")
The reason it works in #2 and not in #1 is the difference in tenses and in verb forms. In "who took my pen" vs "who is wearing my jacket" "who" is the subject in each case, so that's not the difference. (If you took out "who," what would you propose for the subject? Without a subject you have no clause. You might think to claim "John" as the subject, but it can't be the subject of both clauses. You could use a compound predicate: "John is wearing my jacket and is here." )
Notice that in #2 both clauses are present tense. The verb is actually "is wearing," present progressive of "to wear." So you don't actually take out the verb - only the helping verb "is," which leaves you with the present participle, "wearing." You now have a participial phrase, "wearing my jacket," which is just fine. He's wearing it now and he's here now.
In #1, there's no helping verb to take out. You could take out the "who," leaving you with a compound predicate, but you'd need to add a conjunction. "John took my pen and is here." (The two different tenses work, but "is here" is no longer the dominant idea.)
If the original were, "John, who is taking my pulse, is Russian," then it works like #2. "John, taking my pulse, is Russian."
If you use two different tenses, as in the original #1, you'd have, "John, who was taking my pulse, is Russian." That leads to, "John, taking my pulse, is Russian." What happened to your past tense?
So try it, and ask yourself if the meaning is still the same.
Best wishes, - A.