TL;DR: Spell scroll is a consumable item. It holds a spell which can be cast from it or copied into a wizard's spellbook, both of which destroy the scroll. Spell on a scroll refers to that specific spell, which is incidentally written on a scroll.
Spell scrolls (as you'd find in the treasure tables) are spells already prepared onto the scroll and contain some/all of the magic needed to cast them within them (which is why creating them is more than just copying things out of a book).
The second passage you are quoting describes how wizards copy any spell that they find written on a piece of parchment, in a book or on the back of a box of your favourite Orcish breakfast cereal, and put it into their spellbook.
Copying a Spell into the Book. When you find a wizard spell of 1st level or highter, you can add it to your spellbook if its of a level for which you have spell slots and if you can spare the time to decipher and copy it.
Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.
The rules in the DMG are specifically about spells on spell scrolls (the type found in the random treasure tables etc.)
In addition to the rules on copying any spell into your spellbook you also have to follow these rules specific to spell scrolls.
A wizard spell on a spell scroll can be copied just as spells in a spellbook can be copied. When a spell is copied from a spell scroll, the copier must succeed on an Intelligence(Arcana) check with a DC equal to 10 + the spell's level. If the check succeeds, the spell is successfully copied. Whether the check succeeds or fails, the spell scroll is destroyed.
Basically you get one shot to understand enough about this spell scroll before the magic is used up and you can't copy it into your book.
So yes, there's a difference between a plain written spell on any old piece of parchment, and a spell scroll.
Additionally: As KorvinStarmast brought up in the comments you could have someone else help you with this check using the Help action.
Help
You can lend aid to another creature in the completion of a task. When you take the Help action, the creature you aid gains advantage on the next ability check it makes to perform the task you are helping with, provided that it makes the check before the start of your next turn.
Bare in mind that many DMs rule that you can only help with tasks that it makes sense for you to be able to aid someone with. In this instance I would think that at the very least they'd need to have the spell in question on their own class's spell list, or perhaps even be a wizard themselves.
The arcana check seems to be specific to attempts to copy spells from scrolls.
If you read the wizard section of the PHB, in a sidebar they mention you can copy spells from books if you have the required level and the spell is in the wizard's spell list. No check mentioned.
So it probably has more to do with the scroll being a consumable item.
If you were to copy a ritual spell from a scroll, it would be sensible to make you roll an arcana check.
Best Answer
1. No, they don't have a spellbook
Jeremy Crawford (official rules guru for D&D) has said this very clearly:
The only class with a spellbook is the wizard. There is nothing in the Arcane Trickster's description that mentions anything about a spellbook so there is no reason to assume it would get one.
Arcane Trickers may learn wizard spells but they are not wizards.
For example the Arcane Trickster feature says that they know spells:
In essence, the Arcane Trickster knows the few spells so well that they have them always in mind. Knowing spells mean there is no need for a spellbook and no way to use one even if they did have one.
This is directly opposed to way the wizard must prepare their spells from their spellbook.
2. No, they don't have a spellbook
(see 1.)
3. No, they cannot learn spells like a wizard
See this excellent answer for a full discussion.
The Wizard is the only class to have a method to learn new spells aside from level-up or feats. And again the Arcane Trickster is not a wizard in any way.
The Arcane Trickster only lists one way to learn new spells: leveling up.
Note: Arcane Tricksters can use spell scrolls whose spells are on the wizard list
Jeremy Crawford confirmed this:
But they definitely cannot learn spells from them.