Were most castles built by kings or were they also built by nobles?

Yes.

Alan Rufus, First Lord of Richmond, built Richmond Castle.

Roger de Montgomery, First Earl of Shrewsbury, and Earl of Arundel, Sussex, built Arundel Castle.

Walter de Lacy, Lord of Meath, built Ludlow Castle.

Maud de Braose, Lady of Bramber & her spouse, William de Braose, 4th Lord of Bramber, built Pembridge Castle, originally called Newland Castle.

Robert de Crevecoeur, Baron of Chatham, built Leeds Castle.

I can go on. But here, just pull up some of these castles, and can see who built them. Most were built by nobleman who accompanied William the Conqueror, and their descendants.

https://en.wikipedia.org/wiki/List_of_castles_in_E...

And built later by cities to protect the peoples.

Castles in which parts of Europe or the British Isles? The Crusaders (led by nobles and kings, mostly French or English) built several castles in the Holy Land (Middle East in modern terms).

Yes, and lords of the manor, william I built hundreds of castles to establish his rule when he conquered england,he started the tower of london.

They were also built by nobles.

Not all. For example the biggest castle, Caerphilly, was built at great speed specifically because the nobleman (de Clare) who built it - did not have permission. It was a big liability for a King to have unreliable nobles building fortified power bases on the outskirts of their kingdom.

Neither king nor noble ever put a single ounce of their own time and energy into building a castle, instead ordering others to do it, I guarantee you.

Castles were built by whoever needed them. Not all of them were particularly grand or beautiful; they mainly had to be strong and secure. Yes, nobles and even religious military orders like the Knights Hospitallers, Knights Templar, and Teutonic Knights of the Middle Ages also built castles.