This all seems highly confused IMHO, esp. the idea that a mission is secular and a quest is spiritual. As someone who lives in city know for it historical "missions" (Franciscan "posts" for evangelization and other so-called and rightfully controversial "civilizing" activities) and has a school run by "Missionary Oblates of Mary Immaculate" (OMI), plus having seen the film The Mission several times, it seems hard to characterize mission as a secular concept that either Jim Kirk or Ethan Hunt are "sent" on. Cpt. Kirk's is rather open ended and exploratory (more like a journey) while Ethan's highly specific with a very tight time-line (maybe more like an adventure).
I think there is more in common with a quest and mission than difference between those two things and a journey. I think the difference is that to be on a quest is that you can either be sent or initiate yourself whereas a mission is something one is sent by another, be it an authority figure ("King") or even the Divine. A journey is a just a modality of experience, which can be full of purpose from the beginning or can be found in the process of journeying, you may not have to go far to journey either, e.g. Thoreau's cabin at Walden Pond. So in very American fashion, “It's not the destination, it's the journey” which is attributed to Ralph Waldo Emerson and seems to have impacted Jack Keroauc's imagination in his seminal "beat" "travelogue" On the Road, that starts off as a journey but then becomes a quest when Dean Moriarty tells them they are in search of "IT." However, I don't think they are ever on a mission, unlike the Jake and Elwood, aka the Blues Brothers, who are trying to save the orphanage for the Penguin by "getting the band back together."