There is an answer in plain English and there is a loose definition based on the rules.
If you scroll all the way down, my answer for the Goliath is a No. Or at least not without him going through some more hoops.
The plain English answer
You are carrying whatever you and the GM agree you are carrying. For example, by the Merriam-Webster dictionary :
to move while supporting : TRANSPORT
- her legs refused to carry her further
Funnily enough, this could also be argued for other less literal definitions such as:
to convey by direct communication:
- carry tales about a friend
to bear upon or within one:
- is carrying an unborn child
to influence by mental or emotional appeal : SWAY
- She intended the play to carry audiences toward a sense of peace and understanding.
But D&D is usually more literal than this and you will probably get mean looks if you try to argue that you are carry audiences toward a sense of peace and understanding and thus can teleport the whole party with you.
So we can look at how the rules define it.
Carrying as defined by the rules
There is only one rule I've found that really mentions carrying for a player character: it is the Strength attribute definition.
I'll argue that the main definition of carrying is any situation where you would use this rule. Or, looking at it from the character sheet, anything you would list under the Equipment section of your character sheet and count toward your encumbrance limit (even if they have a weight of 0) are things you are carrying. Weather they are objects, creatures or anything else that's managed this way at the table.
It's still not completely defined and the players/GM have some say in what is and is not carried. But it's something.
Lifting and Carrying
Your Strength score determines the amount of weight you can bear. The following terms define what you can lift or carry.
Carrying Capacity. Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry, which is high enough that most characters don't usually have to worry about it.
Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up to twice your carrying capacity (or 30 times your Strength score). While pushing or dragging weight in excess of your carrying capacity, your speed drops to 5 feet.
This includes a few things that are counted as part of your encumbrance :
- The weapon you carry in your hands.
- The armor and clothes you wear.
- Your backpack and whatever is strapped to your backpack or belt.
- The corpse of your friend you bring back to town for reviving..
- Your other friend who is just unconcious but stable that you carry on your back
It's not complete, but we can see highlight a few things that are not carrying based on other rules.
- It's not grappling. While grappling you are holding another character in place by some part of their body, but you are not lifting them off the ground or placing them in your backpack.
- It's not pushing or dragging. As those have specific rules for what happens when your character is above they limit.
- It's not what is inside a cart you are dragging with you, interestingly. As the cart says nothing about this. In comparison, the entry for a horse explains how to handle the weight of a cart.
About the goblin and the goliath?
There is still some uncertainty when it counts to lifting and holding a willing (living) character versus an unwilling character. I think that's just outside the strict definitions given by the book and that's where the GM comes in.
But from what I've laid out, if the goliath just grapples him : No, the goblin is not carried. The goblin is only slightly restrained, as if the goliath was holding it by the arm and preventing him from moving away. If anything, the goblin is being dragged along.
What if the goliath lifted it above the ground? I could see it be argued. But this is not covered anywhere, so you are in GM call territory. Maybe the GM asks for an athletic check to lift it, maybe he asks for a second one to really secure the goblin as being carried instead of merely lifted. Maybe the GM requires the goblin to be tied or restrained or asleep in some way to get a stable handle over it.