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.
No; you can only use slots to cast lower-level spells from your class
The PHB rules (p. 164) for multi-class characters specifically address a multi-class character with slots of a level they can't "know or prepare". And it says You can use those slots, but only to cast your lower-level spells. It further elaborates that if any of those lower-level spells are improved by being used in a higher-level slot, you may get those benefits.
If this loophole worked, a similar loophole would exist for wizards: they can only prepare spells in their spellbook. And the rule for whether they can put a spell into their spellbook is only that the spell must be 'of a level for which [they] have spell slots'. So a multiclass wizard with an up-level slot can find a higher-level spell, and then could copy it into his book, and then prepare and cast it. RAI is pretty clear this won't work (after all, RAW explicitly says it won't, it just also kind of contradicts itself by implication).
In other words, the multi-class character rule saying you can't overrides normal rules for learning and casting spells. It is the "more specific" rule. The character may not 'know' or 'cast' spells higher level than their class levels allow. Which means your example wizard can't copy misty step into their spell book; that would constitute 'knowing' the spell. Nor could they prepare and cast it.
Best Answer
No
The rules on multiclassing and spellcasting state:
In the example above if it is not in your spellbook you can't prepare it as a wizard spell. See the ability arcane initiate. It is designed specifically for the purpose of adding cleric spells to your spellbook for this reason.