What I expected to happen was...:
When pressing F1 all names and the hotbar are hidden.

What actually happened was...:
That hotbar is still hiden and custom names are invisible except for minecart.

Steps to Reproduce (provided by Darth Cobby in this comment):

1. Summon a pig with a visible name:

   /summon pig ~ ~ ~ {CustomName:'"Pig"',CustomNameVisible:1}
   

2. Summon a minecart with a visible name:

   /summon minecart ~ ~ ~ {CustomName:'"Minecart"',CustomNameVisible:1}
   

3. Press F1 to hide the HUD
   →  The pig's name tag is hidden
   →  The minecart's name tag is still visible