[RPG] Can a wizard copy prepared spells into a spellbook

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.

The Player's Handbook on Spells Copied from Another’s Spellbook or a Scroll says

A wizard can also add a spell to her book whenever she encounters one on a magic scroll or in another wizard’s spellbook.… [T]he wizard must first decipher the magical writing…. Next, she must spend a day studying the spell. At the end of the day, she must make a Spellcraft check (DC 15 + spell’s level).… If the check succeeds, the wizard understands the spell and can copy it into her spellbook…. (179)

(Link mine.) So, by the core rules, a wizard can copy any written spell into her spellbook.

However, the Rule's Compendium says, "Spellcasters who use spellbooks can add a spell to their book whenever they find one on a scroll or in another caster’s spellbook. The spell to be copied must be on the copier’s class spell list" (160), and, here, the Rules Compendium agrees with one of the game's designers (see this answer). Ask your DM. This DM has found the game easier to manage for both the DM and the wizard if everyone agrees to limit wizards to scribing into spellbooks only spells that are on the sorcerer/wizard spells list, but another DM's experience may be different.

A typical single-classed wizard can't cast spells that don't appear on the sorcerer/wizard spell list even if they are in the wizard's spellbook

The presence of a spell in a wizard's spellbook doesn't enable that wizard to cast that spell.

The Player's Handbook in the description of the wizard class feature spells says, "A wizard casts arcane spells (the same type of spells available to sorcerers and bards), which are drawn from the sorcerer/wizard spell list…" (56). For example, the typical single-classed wizard just can't cast a spell that's exclusive to the cleric spell list even if the wizard understands that cleric-spell-list-only spell and has it in her spellbook.

In short, a spell must be on the sorcerer/wizard spell list for a wizard to cast it. Alternatively, an individual wizard must uniquely add the spell to her own wizard spell list.

(For example, this DM has always assumed that a creature adds the new spells to the creature's appropriate spell list when the creature succeeds at Researching Original Spells (DMG 198)—as always, bear in mind that researching original spells is vaguely defined, easily abused, and almost entirely at the DM's discretion.)