Yes, and this is explicitly stated on page 114 of the Player's Handbook in the "Your Spellbook" sidebar in the Wizard class description. It works exactly as you are hoping:
If you lose your spellbook, you can use the [procedure in the preceeding paragraph] to transcribe the spells that you have prepared into a new spellbook. Filling out the remainder of your spellbook requires you to find new spells to do so, as normal. For this
reason, many wizards keep backup spellbooks in a safe place.
See the rest of the sidebar for costs and time required.
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.
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 following analysis focuses on spell scrolls in particular, but the same logic applies to spellbooks. In terms of their effect on the number of spells in your spellbook, the two are interchangeable.
In my experience, you won't be a burden
I play a wizard in a game where due to some unfortunate relations with the town militia, my adventuring party and I aren't welcome in town any more. Long story short, me and my DM agreed that since I wouldn't have any access to the components I would need to copy spells into my spellbook in the wilderness, I can't copy over any scrolls into my spellbook for the moment.
I've spent a few levels in this situation (4th to 7th) and it really hasn't been that much of an issue. I feel like I still have plenty of options and never found myself in a situation where I couldn't contribute in some meaningful way.
Granted that's just my personal experience. So let me dive into a few clues in the game's structure that tell us that wizards don't need to get scrolls to be successful.
Even without scrolls, wizards get a huge spell selection.
Wizards start with 6 spells at 1st level, and then gain 2 every level thereafter. Assuming you went pure wizard, by 20th level you will have an absolute minimum of 44 spells in your spellbook that you can prepare from every day. Compare that to other arcane casters: Sorcerers and Warlocks have at most 15 spells known at any given time, Bards have 22 (or 24 if you take the college of lore). And all of those classes start with a fewer number of spells known than you have in your book at 1st level. All of this leads to one conclusion:
By design, wizards have access to more spells than any other arcane class. Only divine spellcasters who can prepare spells off of their entire spell list (clerics, druids, and paladins) can compete in terms of spells available at any given moment.
Spell scrolls are magic items
Magic items, including spell scrolls, are defined in the DMG and the Dungeon Master is given free reign on how much treasure to give out to the party. Under this design, the game has to be balanced in such a way that even without magic items, characters can be successful (since the DM could elect not to give out any magic items at all).
Since spell scrolls are a magic item, and their availability is at the DM's discretion, we can then conclude that the design of the wizard was crafted such that they do not need scrolls to be successful and impactful in the game.