The important point for me about the meanings of TM mantras is that on my TM teacher training course I was directly taught by my course leaders to say that TM mantras have no meaning (in the Fiuggi notes for the 1st lecture and in the 2nd preparatory lecture notes).
The assertion that they have no meaning or religious connotation is/was false. Whether you split hairs as to whether they are hindu gods or vedic dieties is a distinction without a difference. Many, many, many people would be indignant to find out that the mantra they learned in the '70s for 25 or so bucks was religious in nature and derivation. Was in fact the name of Shiva, Lakshmi, Saraswati, or other devas.
Whether you or I would be upset is really moot. The point is the TM movement as represented by its legal agents, that is TM teachers, insisted that the mantras had no meaning. They do. And at least one reported mantra, "ram," is pretty clearly the name of a recognizable god (Rama -- Gandhi on his death uttered "He Ram!" in the belief that by saying the name of God he would be guaranteed eternal bliss).
Look, I've got no problem with uttering "shri ram namah," "shri buddah namah," or "shri allah namah" until the cows come home. And the really old-time TMers who may very well be muttering "shri ram namah" in their meditations may or may not have a problem with it. God bless 'em, enjoy!
But if I were a rabbi/priest/mullah instructed in TM and just now finding out that I've been repeating a sound sacred to a vedic diety for the last so many years -- all the time assuring superiors and parishioners that there was nothing religious about TM, I might very well be fairly pissed and chagrined at this point to find out I had been inadvertently telling an untruth.
I know such religious people. Believe me, they are pissed.
The real question is, why can't the TM movement just say things straightforwardly, outright and simple? For instance, "Yes, in some cultures, the mantras that we use are considered religious. We don't think they have any religious connotations, but our Shankaracharya tradition has found these mantras to be effective. If you are uncomfortable repeating what some religions consider sacred sounds, perhaps TM is not for you. Thank you for your interest. If you don't mind, or even find this interesting, we will be holding instruction on Saturday at 8 a.m. Please bring 3 fruit, 6 flowers, and a new, clean, unused, white handkerchief." I didn't find any such speech in my Fiuggi Lecture notes. Did you? Did any TM teacher?
Why does the TM movement hide behind word games?