Railway terms "Car" is a Sub division of Coach (see below) as in Baggage Car, Buffet Car, Day Car, Kitchen Car, Pullman Car, Restuarant Car, Sleeping Car, etc. Any more?
Note the term Power Car is not correct! Or wasn't until
DMU▸ 's arrived, however it's now in common use as a NEW word.
Coach refers to the Vehicles of a train such as carriages or vans (excluding the Locomotive).
Twenty years ago passenger trains often conveyed Non passenger stock such as parcel vans, even postal vans.
Carriage refers to a passenger carrying coach.
So these days, the term 3 coaches or 3 carriages are both correct!
In days past a 10 coach train may have had typically 8 or 9 passenger carriages.
I surpose the best Modern day example is when
FGW▸ run 9+2
HST▸ 's to Newquay on Summer Saturday with one coach used for luggage it would be correct to say the Train is formed of 9 coaches with 8 (passenger) carriages but that would only confused the public