θρησκεία (threskeia, noun, Strong's # 2356) is used at Acts 26:5; Col 2:18; Jas 1:26, 27 (for those who want to look its use over).
BDAG (Bauer's 3rd edition lexicon, 1995) defines it as, "expression of devotion to transcendent beings, esp. as it expresses itself in cultic rites, worship." AMG's Greek dictionary gives it an infinitive definition, "to worship God."
BDAG also lists θρησκεὺῷ (threskeuo, verb), but this isn't used in the NT.
Both words are derived from θρησκὸς (threskos, adjective, # 2357) which is used in the NT only at Jas 1:26. BDAG defines that as "religious." AMG's Greek dictionary adds "devout" and "pious."
So the NWT's renderings of "form of worship" for 2356 and "formal worshiper" for #2357 ("worshiper of God" for #2357 in the rNWT) fits within the range of meaning given in BDAG.
BDAG is pretty much the gold standard in Greek lexicons. Vine's is a bit dated, but still useful. But I think you are fairly correct in regard to the WT's use of the term "religion." The WT has its own 'religious' vocabulary to which they have nuanced various words. And "religion" and "religious" are often relegated to those outside of WTdom. So their non-use of "religion" or "religious" in a context discussing how Christians should act would seem to fit their MO.
In connection with attention (or lack thereof) given to "orphans and widows," I would posit that here is where the WT's demand for counting field service time is at the root of the problem. It has the effect of forcing people to spend their time where they know it will be counted (counted by the WT, that is), even though the text at James 1:26, 27 shows that caring for "orphans and widows" has a priority with God.
Bobcat