This would refer to armor that was primarily metal. As in, you would look at it and say "that armor is made of metal." As opposed to "That is cloth armor" or "that is leather armor." This is supported by a quotation from the Sage Advice column, where it is talking directly about the Druidic Taboo against wearing metal armor.
The idea is that druids prefer to be protected by animal skins, wood, and other natural materials that aren’t the worked metal that is associated with civilization. Druids don’t lack the ability to wear metal armor. They choose not to wear it.
[...]
A druid typically wears leather, studded leather, or hide armor, and if a druid comes across scale mail made of a material other than metal, the druid might wear it.
http://dnd.wizards.com/articles/features/rules-answers-march-2016
Here, we can see that the metal fittings that would normally be added to studded leather do not count as 'Metal Armor.' And, strictly speaking, you can make Leather Armor with fittings other than metal rivets. I am actually a hobbyist Leatherworker and I do know how to make armor. Ideally, you use metal rivets to hold the pieces together...but it's entirely possible to use stitching, bone, or other hard materials in place of the metal.
For a simple test, just read the description of the armor in the PHB. Every description emphasizes the primary material that the armor is made from. For example...
Studded Leather. Made from tough but flexible leather,
and
Chain Shirt. Made of interlocking metal rings[...]
and as a final interesting point...
Ring Mail. This armor is leather armor with heavy rings sewn into it [...]
In short, while most of this will be DM-fiat, and it's quite possible to have Dragonscale Scale Mail or a Breastplate made from the carapace of some insectoid creature...as a general rule of thumb, if the description of the armor in the PHB explicitly mentions metal, then you can likely assume that piece of armor counts as 'metal' for the purposes of Druids, Spells, etc.
Adventurers League plays by the rules of official books; it does not acknowledge supplements like Unearthed Arcana or Sage Advice.
So the rule from PHB is:
Druids will not wear armor or use shields made of metal
Therefore, per AL, per PHB, no, a druid wearing metal armor would not be allowed by the AL.
Best Answer
While there are no rules in the 5th edition for tiny equipment, you could adapt the rule from the 3rd edition (Player's Handbook, v3.5, p. 123) that armor for a tiny humanoid costs half and weighs a tenth as much as regular armor. Once you've determined a price, the 5th edition crafting rules (Player's Handbook, p. 187) will allow a proficient smith to create it after one day of work for each 5gp value of the armor.
Proficiency for monsters is addressed in the inset box "Armor Weapons, and Tool Proficiencies" on page 9 of the Monster Manual:
The stat blocks for imps and quasits don't specify any equipment, so without DM dispensation they aren't proficient.
It's also the DM's decision as to whether the imp or quasit can fly while wearing armor. Guiding examples:
You're on (or in this case, off) safe ground with light armor, but medium or heavy armor is debatable.