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.

1. No, they don't have a spellbook

Jeremy Crawford (official rules guru for D&D) has said this very clearly:

An Arcane Trickster casts wizard spells but doesn't use a spellbook. The same is true of the Eldritch Knight.

The only class with a spellbook is the wizard. There is nothing in the Arcane Trickster's description that mentions anything about a spellbook so there is no reason to assume it would get one.

Arcane Trickers may learn wizard spells but they are not wizards.

For example the Arcane Trickster feature says that they know spells:

You know three 1st-level wizard spells of your choice...

In essence, the Arcane Trickster knows the few spells so well that they have them always in mind. Knowing spells mean there is no need for a spellbook and no way to use one even if they did have one.

This is directly opposed to way the wizard must prepare their spells from their spellbook.

2. No, they don't have a spellbook

(see 1.)

3. No, they cannot learn spells like a wizard

See this excellent answer for a full discussion.

The Wizard is the only class to have a method to learn new spells aside from level-up or feats. And again the Arcane Trickster is not a wizard in any way.

The Arcane Trickster only lists one way to learn new spells: leveling up.

The Spells Known column of the Arcane Trickster Spellcasting table shows when you learn more wizard spells...For instance, when you reach 7th level in this class, you can learn one new spell of 1st or 2nd level.

Note: Arcane Tricksters can use spell scrolls whose spells are on the wizard list

Jeremy Crawford confirmed this:

The intent is yes, the Arcane Trickster and the Eldritch Knight can use scrolls with wizard spells on them.

But they definitely cannot learn spells from them.