A Mill Deck

Aggro-deck is a term borrowed from games like Magic the Gathering. It means a deck that intends to play aggressively and kill your opponent as quickly as possible.

An aggro deck in Hearthstone is characterized by the following:

• Low mana curve (allows you to play many cards quickly)
• Aggressive creatures (high power relative to toughness) with abilities like charge, battlecry, overload and stealth
• Direct damage (burn) spells

Forcing the opponent to attack your minions with taunt rather than your hero is not a primary characteristic of aggro decks, because an aggro deck will not worry about taking damage since it is going to kill you before you kill them. An aggro deck will typically avoid playing taunt cards because the player is choosing to sacrifice defensive power for offensive strength.

Some cards that would fit the classic "aggro deck" include lava burst, reckless rocketeer, unleash the hounds, and nightblade. These cards fit because they focus on dealing as much damage to the opponent as quickly as possible, and have an effect the turn they enter the battlefield.

The opposite of an aggro deck is a control deck, which is a deck focused on removal and counters, defenses, cards that get stronger over time, and keeping a full hand of cards.

Cards that are neither aggro nor control can be referred to as "midrange" decks. Those focus on cards that are extremely efficient.

At their core, both terms are very similar. There is a slight difference though, which is also touched upon in this reddit comment.

In an aggro deck, every card is supposed to come out quickly and put pressure on the opponent. The term is very well explained in the question you linked, so I won't get too specific, but what I want to emphasize is that the aggressive nature of the deck means every card supports the aggressive mindset and tries to do something immediately.

Rush decks, on the other hand, are defined by their goal of winning quickly. While this usually means being aggressive (in fact, most rush decks also qualify as aggro decks and vice-versa), it can sometimes mean trying to get to a cheap combo quickly. The important point here is to win as quickly as possible, giving the opponent a minimum of time to setup some sort of big strategy.
Current Hunter decks are an example, while they try to win quickly and put the opponent on a clock, they don't necessarily do so aggressively by playing lots of minions and attacking, but can sometimes wait a few turns just doing damage with their hero power and Eaglehorn Bow, using Secrets to keep the Bow on their field, waiting for everything to fall into place for their big Unleash the Hounds combo.

Basically, "rush" refers to a kind of time frame (early/mid/late game), the others being midrange and control, while "aggro" refers to a strategy (e.g. rush, combo, aoe, big guys, ...).

So, to summarize:

Aggro decks

• have an aggressive playstyle
• use every card as quickly as possible
• put pressure on the opponent at all times
• can choose to trade favorably instead of doing as much damage as possible

Rush decks

• try to win quickly
• sometimes hold on to cards and wait for combos
• put the opponent on a clock
• nearly always do as much damage as possible