Seems like a lot of the confusion is caused by people who don't read.
Stop and think about it. If you are flying out, and you are Temp or Perm, when you get to the airport you don't have the bottom portion of the FMM form. So you go to INM, complete the form, the agent stamps it and gives you the bottom portion so you can hand it to the ticket agent and they can clip it to your boarding pass. That's pretty straight forward.
When you return, the video describes exactly what you need to do. The reason that they are emphasizing this so much is because when people return, and they don't do it properly they run the risk of voiding their Temp or Perm and having to start the whole process over again. As they tell you in the video, complete the FMM, hand your Temp or Perm visa to the INM agent 1st, then give him your passport and 3rd give him the FMM with only the top portion completed. Only people coming in on a Tourist visa get the bottom portion returned to them.
Just watch the video.