[RPG] Can a multiclass character with 1 level of wizard copy any wizard spell they find into their spellbook

First off, there's going to be a lot of variation here. Not all adventures have opportunities to access a spellbook, so you're likely to see whole campaigns where it never comes up and others where they're everywhere.

Lets look at some examples though, from various seasons of Adventurers League, just out of curiosity.

Season 1: 6 spell scrolls from the tier 1 adventures, most of which are on the Wizard spell list, and thus could theoretically be copied into a spellbook. In the 5 tier 2 adventures every single one of them has a spellbook or two that the players can theoretically get access to. There are also several scrolls. Apparently the authors went spellbook happy in the Tier 2 parts of this season.

Season 2: The tier 1 adventures have a handful of scrolls and 2 spellbooks. The Tier 2 adventures have a handful of scrolls and 1 spellbook.

Season 3: Tier 1 has a handful of scrolls and no spellbooks. Tier 2 has a few scrolls and 2 spellbooks. Tier 3 has a few scrolls and 1 spellbook.

In each of these cases the spellbooks in question have a number of spells, some of which you could immediately copy and some of which you need to wait till you are higher level. It's also not necessarily certain that the players will get access to the spellbooks. There's also some serious overlap between stuff in each spellbook and stuff the PCs probably already know.

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.