Is wanting to do something = desire?

Is wanting to do something = desire ?

Not really. Wanting to do something is a thought process. In Buddhism, desire (Taṇhā) refers to greed or craving pleasure, material goods, and immortality, all of which are wants that can never be satisfied as the world is impermanent in nature and it is always subjected to change. This causes suffering. People generally desire a better life, family, children, money, cars, boats, planes, vacations etc. All these things which are impermanent.

Enlightened people do not possess material craving. Followers of Buddhism tries to mitigate these desires and understand that everything is impermanent. Yes, they do feel hunger, they do feel pain and pleasure but they understand that both pleasures and sufferings are impermanent. For example, a normal person might crave luxurious food while an enlightened person might only eat food just to survive and stay alive.

"Monks have intention to teach dhamma to others"

The monks intention to teach Dhamma is not done for the pleasure or for greed of spreading the religion. If I am not mistaken, I think there are some guidelines for monks when leading an ascetic life and some of them do require a monk to teach so more people open their eyes.

How a enlightened monk differs from an enlightened lay person ?

A monk would abide by the guides related to being a monk such as preaching Dhamma and any other responsibilities set around a monkhood. Lay person is not given these rules hence might live a quiet life. Either way both have understood the impermanence taught in Buddhism and will have no material craving. If by enlightment you mean attained nirvana, they will no longer have another life. ( Remember there are stages of enlightenment. More explained here.

Is wanting to do something = desire ?

Abhidhamma...

1. Thinking to do can be wholesome, unwholesome, or neither wholesome nor unwholesome.
2. Thinking to attach can be unwholesome only.

When one thinks to eat, he can eat by wholesome, unwholesome, or neither wholesome nor unwholesome. His mind is unwholesome by desiring in eating, wanting to eat more and more without proper reason.

Doesn't enlightened people want to do anything ? They just react to whatever comes ? ( "react" may not be a suitable word).

The Arahant want to do the best for people without attachment. The enlightened people are not a robot. They react by developed wisdom. The developed wisdom is automatic, but it is adaptable for peoples' 6 advantages. The robots can't act like that because they still are programmed.

A related question that I hope to ask in Buddhism SE: How an enlightened monk differs from an enlightened lay person?

They are the same, but Vinaya Rules are forcing all monks, either enlightened or not, to teach in the way leading Dhamma to 5,000 years. In Vinaya's commentary commented "The Arahants' job is reciting, memorizing, and understanding whole Tipitaka." Every enlighten people should teach and memorize Tipitaka. Vinaya and commentary allowed even the ordinary in reciting and teaching the Dhamma, but the Vinaya and it's commentary focus on Arahanta and all Buddhist Teachers, either enlightened or not, in memorizing Tipitaka because they has no left meditation for enlightenment to do. That's why the Buddha denied the monks to keep foods for eating, the Buddha forced the Arahant to meet people and teach them.

An enlightened mind has eradicated the fetters that bound that mind with wandering around with suffering born from ignorance and craving.

What kind of craving?

In the Discourse of the Setting in Motion the Wheel of Dhamma (SN 56.11), the Buddha said this about the kind of desires than bind to suffering and dissatisfaction:

"And this, monks, is the noble truth of the origination of stress: the craving that makes for further becoming — accompanied by passion & delight, relishing now here & now there — i.e., craving for sensual pleasure, craving for becoming, craving for non-becoming.

These three kinds of craving are called Tanha, or 'thirst'. As the objects desired from that thirst are impermanent, the emotional and mental thirst is never quenched by getting this or that object.

Other kinds of desire may still arise for the enlightened mind. The main difference is that such kinds do not lead to suffering, because ignorant expectations, assumptions and generalizations about the world have been uprooted. For such liberated mind, the task has been done. There is nothing further for the sake of this world.

Kind regards!