Yes, but only 1st-level spells
PurpleVermont has already argued why such a multiclass could only write down spells of a level they could cast as a Wizard (so actually up to 2nd level, if they take Wizard 3). I disagree with Vermont's argument that stolen spells could not be transcribed at all, however.
The rules don't clearly state that you have to have seen a spell written down to be able to transcribe it, that's a matter of interpretation--and I disagree with the interpretation:
Copying a spell into your spellbook involves reproducing the basic
form of the spell, then deciphering the unique system of notation used
by the wizard who wrote it. You must practice the spell until you
understand the sounds or gestures required, then transcribe it into
your spellbook using your own notation.
The fact that different wizards use different notation implies that the written form of the spell is not an inherent part of the magic, merely an aide-memoire. And the requirement to practice the spell until you understand its V/S requirements -- precisely the thing which the Arcane Trickster can instantly intuit -- strongly suggests that this is the most important part of "learning" a spell, and the rest is merely paperwork that your first-level Wizard training would absolutely prepare you for.
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
From the "Your Spellbook" sidebar, page 32 of the Player's Basic Rules (v0.2):
Emphasis mine. Only spells on the Wizard list can be learned in this way. In this area, there is no difference between spells written in a spellbook and spells written on scrolls. The difference is between Wizard spells and non-Wizard spells.
Even if the game allowed you to copy a non-Wizard spell into your spellbook, we have in Preparing and Casting Spells, page 30:
So even if you could write non-Wizard spells in your spellbook, you couldn't prepare them and therefore couldn't cast them.
Further down the same page, you can cast rituals without preparing them, but even there, we have:
So, you can't copy non-Wizard spells into your spellbook, and even if you could, there's just no way to cast them.
Wizards can't use scrolls that are not on their list either, since we have on page 60 of the DM's Basic Rules:
As to why this is (from a narrative perspective), every method of spellcasting works differently. For example, the Wizard and the Sorcerer probably have the most similar spell lists, but the Wizard uses magic through painstaking study and preparation, where a sorcerer simply unleashes the power they have within them. So a spell that isn't on the Wizard spell list is probably just impossible to cast using the Wizard method of spellcasting.
From a balance perspective, obviously it would be completely unfair if Wizards had access to every spell in the game.