I did find a few references for training coat, and you certainly would put that over a training jacket. I think this sounds better than putting a warm-up jacket over a training jacket, but there are lots of examples of warm-up jackets that certainly could be worn over another jacket.

Without seeing a picture of the original it's all a lot of guesswork though.

uhlsport Classic Training Coat
Long cut training coat Features: - 1000 mm coating - taped seams - armcuffs 1:1 rib - detachable hood - zip pockets - inside chest pocket Material: Shell: ...
www.uhlsport.com/product/germany/en/3982 - 6k - Cached - Similar pages - Note this
uhlsport classic training coat
Veste 3/4 - coutures plates - manches 1:1 rib - capuche détachable - poches zippées - poche intérieure Matière : Extérieur : 100% PES Doublure : 100% PES.
www.uhlsport.de/product/france/fr/3982 - 6k - Cached - Similar pages - Note this
LIVERPOOL FC TRAINING COAT on eBay English League Clubs, Football ...
Find LIVERPOOL FC TRAINING COAT in the Sporting Goods , Football Shirts , English League Clubs , Liverpool category on eBay.co.uk.
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JUVENTUS DEL PIERO PLAYER TRAINING COAT on eBay Overseas Clubs ...
Find JUVENTUS DEL PIERO PLAYER TRAINING COAT in the Sporting Goods , Football Shirts , Overseas Clubs , Italian Clubs category on eBay.co.uk.
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Nags Training Coat:Wind-breaker Slazenger: Windcheater, 100 ...
Wind-breaker Slazenger: The fashionable 100% ripstop nylon wind-breaker from Slazenger has a shiny silver coated inner and a black mesh lining.
www.spreadshirt.net/shop.php?op=article&article_id=3765062&... - 14k - Cached - Similar pages - Note this