It mostly seems that 'luge' is considered a small type of "traineau". Both Larousse and Robert indicate that in their definitions as well.

https://dictionnaire.lerobert.com/definition/luge

Quote from Larousse sums this up : 

luge

nom féminin

(mot savoyard, du bas latin sludia)

To discuss the difference in meaning between 2 words, you need the definitions of both words.

A luge is a ‘petit’ traîneau. Not all traîneaux are luges. 

A sled-like snow vehicle is also a traîneau, and is referred to as ‘un traîneau’ in French, but it is not the only traîneau.

3 significant French monolingual dictionaries - Robert, Larousse and Académie-Française confirm this (review both words for completeness). TV5, amongst many others, agrees : 

https://langue-francaise.tv5monde.com/decouvrir/dictionnaire/l/luge

A simple internet search will confirm the description of a luge also belonging to the ‘family’ of traîneaux - Amazon France being just one of many examples

https://www.amazon.fr/Luges-traîneaux-50-à-100-EUR/s? 

Wordreference also defines a luge as a (petit) traîneau. https://www.wordreference.com/fren/luge

Wordreference is a good word to word, and fixed expression translator - in many ways it is more like a bilingual thesaurus than a reference dictionary though. 

For comprehensive definitions of words that include the nuance and detail, a good monolingual reference dictionary is superior. Luckily, we are studying language in an era when there are numerous comprehensive, reputable (French) dictionaries accessible on line.