Short Answer
'his spellbook' should be changed to something more sensible, like "any of their spellbooks" or "any spellbook on their person". The rules expect, for absolutely no good reason, that each wizard has exactly one spellbook and they carry it on their person at all times except for rare occasions when they are securing it hidden away in some variety of magically protected vault. Magically protected vault think and on person think don't seem to be aware of each other and on person think definitely dominates.
It's an easy fix, though-- you can even fix it just by getting creative with your definition of 'his spellbook' and claim that the RAW kinda works1
RAW Silliness
'his spellbook' is used several other times in the text. Let's look at what it means:
A wizard may know any number of spells. He must choose and prepare his spells ahead of time by getting 8 hours of sleep and spending 1 hour studying his spellbook. While studying, the wizard decides which spells to prepare.
A wizard begins play with a spellbook containing...
(emphasis added)
At each new wizard level, he gains two new spells of any spell level or levels that he can cast (based on his new wizard level) for his spellbook. At any time, a wizard can also add spells found in other wizards’ spellbooks to his own (see Magic).
Spells Gained at a New Level: Wizards perform a certain amount of spell research between adventures. Each time a character attains a new wizard level, he gains two spells of his choice to add to his spellbook. The two free spells must be of spell levels he can cast.
A wizard must study his spellbook each day to prepare his spells. He cannot prepare any spell not recorded in his spellbook, except for read magic, which all wizards can prepare from memory.
Wizards perform a certain amount of spell research between adventures. Each time a character attains a new wizard level, he gains two spells of his choice to add to his spellbook.
Wizards can add new spells to their spellbooks through several methods.
A wizard can also add a spell to his book whenever he encounters one on a magic scroll or in another wizard’s spellbook.
If the bearer is a caster who prepares arcane spells, he may prepare this spell as if it were in his spellbook.
(Wayfinders)
However, this would allow a wizard to copy a spell into his spellbook so he could prepare the spell from his book (assuming the spell takes up only one page...
(Memorize Page)
So, whatever 'his spellbook' is, it's the same 'his spellbook' that the Wizard has to study to prepare spells and that spells are instantly poofed into when the wizard levels up.
The spellbook a wizard starts with may or may not be 'his spellbook'; the rules refer to it only as a spellbook, much like those that can be purchased from the equipment section of the rulebook. Nonetheless, an entity need not be physically present to be studied, and adding spells to 'his spellbook' is possible both through levelling up and through the spell Memorize Page.
Since a wizard's spell are all in 'his spellbook' and a wizard preps by studying 'his spellbook' and adds spells to 'his spellbook' all without ever needing to physically interact with 'his spellbook' all you're doing by exploring the RAW wonkiness here is buffing wizards by making 'his spellbook' harder to destroy/endager/take away. That's... not really helpful for anything ever. Unless you're a wizard trying to protect 'his spellbook', which is currently a bunch of flamable paper in a backpack, in which case convincing your DM that 'his spellbook' is actually merely the platonic idea of a spellbook with no special relation to that specific physical copy may in fact be useful.
In any case, the RAW never specify what 'his spellbook' means, specifically, nor do they specify what ownership, possession, or 'his' means. Spellbook is relatively clear. Since the rules do not define these terms they are free to be defined however the GM sees fit, provided those definitions are compatible with the RAW.
1 Specifically, define a spellbook broadly-- a wizard's spellbook is the thing they stores their spells in. If they store their spells in several books it is the collection rather than any individual book that is 'his spellbook'; the books merely serve as component parts. You can additionally specify that ownership is possession, and more specifically physical possession, so that books in the wizard's personal library back at their estate don't count as 'his'. While this works (the second order consequences for magic items that rely on ownership and such are very much in line with the theme imposed on the wizard by this limitation), it's definitely gymnastic and it's probably simpler just to replace the rule with an equivalent and more clear-on-its-face houserule.
Oh, and obviously the 'his' in his spellbook is gender neutral.
No you cannot transfer spells prepared as a Cleric into your spellbook
Just above the text that you reference, the sidebar in the PHB explains
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare
the time to decipher and copy it.
The fact that the book explicitly says that it contains wizard spells is the killer here. Spells that you prepare as a cleric are not wizard spells. They are cleric spells, even if they are on the same spell list.
We know this because of the rules for multiclass spellcasters (PHB pg. 164)
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell. Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.
From this, we establish that your prepared spells as a cleric are "Cleric Spells" in that they are prepared as a cleric, and cast as a cleric; not as a wizard. And your prepared spells from your spellbook are "wizard spells" for a similar reason. Even if the two appear on the same spell list, each spell you have prepared is associated only with one of your classes.
The PHB then goes onto explain that when copying spells from spellbook to spellbook:
This is just like copying a new spell into your spellbook, but faster and easier, since you understand your own notation and already know how to cast the spell.
If copying from book to book is the same as adding new spells except faster, then we know that you can only transfer over wizard spells since you can only copy new wizard spells into the spellbook.
Finally, As you have pointed out:
you can use the same procedure to transcribe the spells that you have prepared into a new spellbook
The procedure in question is the same as copying over a new spell, or a spell from one book to another. Both of these processes require wizard spells to work. Thus, even though copying a spell down from memory doesn't specifically add any new restriction, it doesn't explicitly lift the general limitation on the procedure; the precondition that the spell in question is a wizard spell.
Best Answer
A magus who prepares a spell using the class feature knowledge pool can copy that prepared spell into a spellbook—probably
That is, general consensus seems to be a tentative Yes in this 2011 Paizo messageboard thread. Developer Jason Bulmahn solicited opinions during the magus playtest as to whether this should be considered a bug or feature in this 2010 Paizo messageboard thread.
There is some discussion whether the magus technically understands the spell—a necessity when copying a spell from another caster's spellbook or from a scroll—but there's no allowance made for not understanding a spell that's already prepared, the assumption likely being if it's in your head, you understand it.
No one points to anything horribly unbalanced about this quirk, but several do voice that allowing the magus to do this makes them vaguely uncomfortable, but only insofar as such a process goes unmentioned in the magus's description.