Does Facebook ad click mean that they clicked on your mobile video ad or that they clicked on install and went to your app?

The generic click action means a click anywhere on your ad. On the app, on the video or even on the page name. To drill down you should use Facebook Ads reporting tool where you can add columns to display custom actions like app installs and many more (assuming you have Facebook Sdk correctly setup). You may also want to use an External optimization/management tool (like AdEspresso or Qwaya) once you cross the $2000/mo in spending. They'll save you a lot of time and money.

You should be able to find out this info in Facebook's reports. Go to the campaign you're running in the Ad Manager and click on the View Report button. As soon as you're taken to the Reports page, go to Edit Columns. In there, go to Actions and then select all the actions (clicks to play video, video views, app installs, etc) that you want to see. If you want to see the cost per each of those actions, also check those same boxes under Cost per Action. I also like to choose one of the options under Data Breakdowns to see who's giving me the most/cheapest cost per specific action. When you've selected everything you want to see (and de-selected everything else), click Save Columns. Once you've got your columns looking the way you want, make sure you save the report. That way you can see this info whenever you want without always having to adjust your columns. Give that a try and let me know if you're able to see everything you were looking for!