Which verb is used to describe a person who is responsible and organising for doing something?

A verb that can take on the several meanings expressed in the question is engineer:

[Merriam-Webster]
1 : to lay out, construct, or manage as an engineer
// engineer a bridge
2 a : to contrive or plan out usually with more or less subtle skill and craft
// engineer a business deal
2 b : to guide the course of
// engineer a rally

Somebody who engineers something is generally somebody who plans or initiates it, and is often also the person who oversees it to completion.

---

Using this word, the sentence in the question would become the following:

He is engineering this website.

In that sentence, as with most involving engineer, there is no need to also use the verb build; engineer includes the sense of build, so adding it would be redundant.

As in:

He authored this website.

to author verb TFD

To write or construct (an electronic document or system): authored the company's website.

An author (v. a writer) is an originator or creator, involved in a piece of a work from start to finish and generating the ideas behind it.

He is in charge of building the website. He heads the building of the website.

These do not imply that he initiated the project, only that he is its manager.

The website is his baby.

This is a joking way of saying that he conceived it, he is in charge of the project, and he might react irrationally if the project is threatened.

Would Orchestrated fit the bill?