Identifying a scroll is different from deciphering a scroll and neither requires expending the scroll.
Identifying
The detect magic spell determines if there're any magic auras in the area on the first round, the number of magic auras and the most potent aura in the area on the second, and where exactly those auras are and how powerful each is on the third. With a successful Knowledge (arcana) skill check (DC 15 + spell level), the caster can determine the aura's school of magic; this check apparently takes no time and is part of the 3rd-round detect magic effect. Doing this would tell the caster only the school of the spell or spells on the scroll.
A 3-round thorough examination of the object while employing the detect magic spell permits the caster to "[i]dentify the properties of a magic item" with a successful Spellcraft check (DC 15 + scroll's caster level). This tells the caster everything about the magic item, with apparently one exception. The caster knows whether the scroll is arcane or divine, what level spell is on it, what level the spell or spells are scribed at, and so on, but identifying doesn't reveal exactly what spell is on the scroll; to do that the caster must decipher the scroll.
These 3 rounds are in addition to the first 3 rounds leading to pinpointing each aura in the area of the spell detect magic. Events happen in order unless stated to happen simultaneously.
(The analyze dweomer spell also reveals these properties, but the word dweomer is difficult to say without laughing.)
Deciphering
To find out which spell is on the scroll (and subsequently be able to activate it) the caster must decipher the scroll, which requires a successful full-round Spellcraft skill check (DC 20 + spell level), a successful 1-minute Use Magic Device skill check (DC 25 + spell level), or the read magic spell.
After the scroll's deciphered, that scroll's always deciphered for that caster. No further checks need be made, and the scroll can be employed normally.
Don't conflate identifying with deciphering. The hairs are fine and difficult to split but the game splits them anyway: When a caster identifies an item's properties, he gets command words, number of charges, and so on (pretty much just like he read the item's description from the book) but not the name of the spell on the scroll; the caster gets that and the ability to activate the scroll from deciphering the scroll.
Note: That's weird and the Pathfinder Role-playing Game buries that on page 490: "The writing on a scroll must be deciphered before a character can... know exactly what spell it contains." This also isn't exclusive to Pathfinder--both the Dungeons and Dragons 3.0 and 3.5 Dungeon Master's Guides have the exact same language in their Deciphering a Scroll sections (pages 203 and 238, respectively).
Note: A solid but dangerous case can be made for skipping the identification step if the caster already knows what he's dealing with, much in the same way someone proficient with a sword can just whack fools with the sword instead of learning its maker, properties, and history before he starts using it. The DM rolls secretly to determine if a character deciphers gibberish via the Linguistics skill, but the player rolls openly to decipher a magic scroll via the Spellcraft skill or Use Magic Device skill, so right there the player will know it's a magic scroll and, if successful in deciphering it, know the spell on it. The character can then try to activate the scroll, but--and this is really a DM's call--the character still won't know any of the scroll's properties (such as it's spell level or the scriber's caster level) because the character's not identified the scroll's properties. Just as Fighter Joe can still take swings with a magic sword without knowing its exact properties, Wizard Jim can decipher a scroll and, if he meets the requirements, cast the spell from the scroll without knowing exactly what's going to happen. I've never seen a player want to do that, but I'd totally let him. That could be hilarious.
Other Tidbits: There are spells that straight-up answer questions and get information; one could possibly use those and make a big pile of information about a scroll. There are feats and classes that are scroll-specific, and those might be useful. I've ignored both because they seem to be beyond the question's scope.
Sorcerers can use a scroll of a Sorcerer/Wizard spell they do not know.
The "If he can't already cast the spell" line that causes you confusion is more clear in context:
To use a spell completion item safely, a character must be of high enough level in the right class to cast the spell already. If he can't already cast the spell, there's a chance he'll make a mistake.
It refers to the Sorcerer's class level, the character's potential to learn and cast such a spell.
The clause you emphasized in the second quote, "The user must have the spell on her class list", reinforces this. It mentions that the spell must be on the correct list (in this case, the Sor/Wiz) list. It need not be on that Sorcerer's list of spells known - that would be a more stringent condition. Luckily for Sorcerers, that's not the case.
And yes, the same applies to Bards.
Best Answer
There are other arcane spellcasting classes that use spellbooks, like the magus. The alchemist has a “formula book” that works mostly the same way, and the witch has a familiar that functions like a spellbook for her.
In all of these cases, however, learning from scrolls is reserved for a class with very wizard-like spellcasting, that is, they must prepare exactly which and how many of each spell they know in advance.
As you note, the divine spellcasters who, like the wizard, prepare spells ahead of time don’t need to bother with spellbooks at all. Clerics, druids, and similar divine spellcasters can pray for any spell on their list, even one they have never seen before. In effect, you can think of the gods they pray to as being giant spellbooks that already have every spell available ready to go.
On the other side of the coin, we have spellcasters that do not have to prepare their spells ahead of time. The sorcerer is the quintessential example, but others exist, including some like the oracle that are divine (as you note). These “spontaneous” classes do not keep a spellbook or similar, and they cannot take advantage of the rules for copying a scroll into a spellbook. Scrolls, for them, can only be used as a scroll, not used to learn a spell. To learn a spell, they need the vastly more expensive page of spell knowledge, or the even more expensive ring of spell knowledge (which then does allow them limited ability to learn spells from scrolls and other sources).
The advantage to spontaneous spellcasters is that they are free to use these spells in any combination they like, up to their spells per day limit, where spellcasters that prepare have to choose exactly how many of each spell they want. However, generally speaking, the advantage of having access to so many spells is vastly superior, so clerics and druids and wizards are generally better than oracles and sorcerers and the like.