The correct word is "initiated". Both DILD and WILD were coined by dream researcher Dr. Stephen La Berge in his book, "Exploring the World of Lucid Dreaming". 
 

Wake Initiated Lucid Dreams are dreams that initiate (or start) at the point of waking, and the dreamer falls asleep consciously. 

Dream Initiated Lucid Dreams are dreams that initiate (or start) during a dream, it is where the dreamer gains critical awareness during a dream. 

A common mistake is to believe the "I" stands for "induced", however, induced means "caused by".

Wakefulness does not cause lucid dreams any more than non-lucid dreaming causes lucid dreams. Therefore using the incorrect "induced" makes these names rather meaningless and misleading.