The "Attack action" is a specific action; attacks are anything with an attack roll.
What is the Attack action?
The "Actions in Combat" section of the Player's Handbook (PHB, page 192) states (emphasis mine):
When you take your action on your turn, you can take one of the actions presented here, an action you gained from your class or a special feature, or an action that you improvise [...]
The use of "or" clearly means that class features and the "action listed here" are distinct actions. The first action listed is "Attack"; the "Attack action" (emphasis mine):
The most common action to take in combat is the Attack action [...] With this action, you make one melee or ranged attack.
This, and the above quote, clearly show that the Attack action and attacks are different things. There is only one Attack action, which is why it is defined alongside the other actions such as Dodge, and Hide.
What is an attack?
Regular attacks are defined in the "Making an Attack" section (PHB, page 194):
If there’s ever any question whether something you’re doing counts as an attack, the rule is simple: if you’re making an attack roll, you’re making an attack.
They are different things
Bringing all this together: the Attack action is a specific action that anybody can take whereas class granted features and abilities, such as the Ranger's Volley, are not Attack actions, but instead their own individual actions even though they may happen to involve attack rolls.
In conclusion: There is the "Attack action", which is a specific action and there are "attacks", which are anything involving an attack roll. The easiest way to tell these apart is that one is followed by the word "action" and the other is not.
Capitalization is imperfect, but the phrases are entirely different
As Nitsua60 correctly points out in his answer:
A close reading of the PHB makes clear this is a place where the authors have been very consistent and explicit in differentiating: you'll always see phrases like "make an attack" and "take the Attack action," never "make an Attack" or "take the attack action." See PHB pp. 49, 63, 72, 74, 78, 79, 85, 92, 93, 111, 165, 168, 170, 192, 195, for some examples of their consistent and exacting capitalization." (see their answer below for more details)
Icyfire also correctly points out in their answer that capitalization isn't perfect; the best way to distinguish between these terms is simply by whether or not the word "attack" is followed by the word "action" (see their answer for more details).
Another reason capitalization isn't perfect is that many sites, including Roll20, completely butcher the capitalization of the actual rulebooks by capitalizing countless words that were not capitalized originally. This means capitalization is not something you should rely on for clarity, and instead you should go by whether it says "an attack" or "the attack action".
Miscellaneous further justifications and clarifications
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn....
This helps to showcase the difference between the two. If "attack" and "Attack action" were synonymous and thus making an attack meant taking the Attack action, then the Feature could simply read "You can attack twice, instead of once, on your turn."
This type of reading would also lead to allowing for infinite attacks in a turn - clearly not the intended reading.
Note: This feature is not an action nor attack of its own, but a passive feature which activates "whenever you take the Attack action on your turn".
Also Note: The reason "Extra Attack" is capitalized is the same reason any feature would be capitalized: because it is a specific/proper phrase/noun.
When you take the Attack action and attack with a light melee weapon...
Again if "Attack action" and "attack" were synonymous then this sentence is redundant and could be reduced to "When you attack with a light melee weapon", the fact that it is not reduced further shows that they are two different things.
In order to cast most spells you must use your action to take the "Cast A Spell" action. If "attack" and "Attack action" were the same thing and you were casting, say, fire bolt, a spell which has an attack roll, then you would have to use the "Cast A Spell" action but then you would have to take the "Attack" action as well (as you are making an attack). This is clearly impossible as you, generally, only have one action per turn.
Thus we can conclude that something having an attack roll does not make it require the "Attack action" or else a consequence would be the inability to cast spells with attack rolls.
[...] you can use the Attack action to make a special melee attack, a grapple. [...]
Using the Attack action, you can make a special melee attack to shove a creature [...]
These are attacks made using the Attack action and are specifically called out in the rules as being "special". They are special for two reasons. First, they do not involve attack rolls, but are still considered attacks. And second, because they replace attacks made as part of the Attack action, and not just any attack. The best example of this is opportunity attacks, which cannot be a grapple or a shove as you are not taking the Attack action but instead a reaction that involves an attack.