The information conveyed is left up to the DM.
By looking up the way Arcana checks are described in the Player's Handbook, it doesn't need to be an excerpt out of a spellbook. The kind of knowledge coming from identifying a spell could be the spell's name with its full description, could be a vague interpretation of its effects, or some folklore about a witch using that spell in a nursery rhyme.
Arcana. Your Intelligence (Arcana) check measures your ability to recall lore about spells, magic items, eldritch symbols, magical traditions, the planes of existence, and the inhabitants of those planes.
(that's the most there is in term of rules about character knowledge)
Of course, identifying a spell that is being cast is more about theory and practice than trying to identify a spell after having seen its effects. As the DM, you are the one deciding what information is conveyed by identifying a spell. To a wizard it would be more likely to give a detailed description of some sorts of the spell given how much time they tend to spend studying, but it doesn't need to be an excerpt out of a spell encyclopedia. It could also be that the wizard (or any other character) does not know the spell but is theorizing about it by making some assumptions using what they know about magic.
It might be worth mentioning that the spell identify can reveal in more detail what spells are affecting a creature or object, if any.
Remember that this kind of spell-identification requires that the character has perceived the casting, detailed in the paragraph before the one you are quoting.
Many spells create obvious effects: explosions of fire, walls of ice, teleportation, and the like. Other spells, such as charm person, display no visible, audible, or otherwise perceptible sign of their effects, and could easily go unnoticed by someone unaffected by them. [...]
And I might highlight the important part from your quote for this purpose.
If the character perceived the casting, the spell's effect, or both, the character can make an Intelligence (Arcana) check with the reaction or action.
In your example about "Bob acting strangely lately" it might be impossible identifying the spell affecting Bob if you cannot perceive it as a spell. You can theorize Bob is under a spell, and that he has been acting strangely because of that, but you don't know for sure if Bob is actually under a spell or just a bit under the weather. Trying to identify such an unperceivable spell would be theorizing about that it might be, and the nursery tale / folklore aspect of the description might be more fitting in this case than a spell's name or description.
Can a character successful at identifying a spell place it on a spell list?
Spell lists tend to be there mostly for player's mechanics. You can assume only warlocks can cast armor of Agathys, but what if the one casting it is a dwarf with the Mark of Warding Dragonmark (which adds that spell to any of their classes spell list)? Or what if that spell is just innate magic, like the spells gained by playable races such as tieflings and drow or many other monsters? Or perhaps some other kind of special feature, like the Magic Initiate feat.
There are also other exceptions to spell lists in the form of expanded spell lists (for warlocks) or extra spells (for cleric domains, druid circles, or paladin oaths), which add new spells to that character's class spell list.
does this imply RAW that any character of Int 10 potentially has encyclopedic knowledge of every spell of up to 5th level?
Yes and no; you are – as the DM – the one deciding what kind of information a character gets. If a character with Intelligence 10 and no Arcana proficiency does indeed succeed at their check, you can probably tell them about stories they might have heard as a child or some rumors they thought were too imaginative before that moment. It could even happen that such a character has managed to identify the spell when a fully proficient wizard didn't, and you can say that the wizard was being too methodical in their analysis.
Remember that the rules in Xanathar's Guide to Everything are optional and that you can use them however you prefer. If for you identifying a spell with that rule should give a full description, you are allowed to do so. It's been left vague enough for this reason.
From the introductory paragraph in Chapter 2:
The material in this chapter is meant to make your life easier. Ignore anything you find here that doesn't help you, and don't hesitate to customize the things that you do use. The game's rules exist to serve you and the games you run. As always, make them your own.
