Maxperson
Morkus from Orkus
No, a dex check is not about "remaining still and silent". A dex check is about doing something which allows you to either conceal your position or refrain from revealing it. It might be remaining still, but you can make hide checks when moving in some cases.
The problem here is that you invented a shiny new rule that is not part of the game. READ THE ACTUAL RULES.
Stealth. Make a Dexterity (Stealth) check when you attempt to conceal yourself from enemies, slink past guards, slip away without being noticed, or sneak up on someone without being seen or heard.
"An invisible creature can't be seen, so it can always try to hide. Signs of its passage might still be noticed, however, and it still has to stay quiet." => you can hide WHILE MOVING, but "signs of your passage" (such as footprints) could be noticed.
Note that "give away your position" may not entirely invalidate hiding. Consider: You turn invisible. You shout. You have now "given away your position". Can you try to hide, without moving? You can! An invisible creature can always try to hide. The mere fact that everyone knows where you were before you hid doesn't mean that they can tell you haven't moved.
In a game full of rulings-not-rules, you've latched onto the phrase "give away your position" and built a giant edifice of dreams and speculation on it, without considering that it might not be intended as an absolute definition of an essential and inevitable component of stealth, which must be used to redefine every other term or statement made.
He has also consistently ignored that the rules say that the DM has the option to use other stats for ability checks, so by RAW, he could use WIS or INT for the wood elf hide ability. I'm thinking it was ignored because it neatly defeats his reasoning that RAI doesn't make sense, because hiding while someone is looking at you shouldn't be a DEX check in his opinion.