Eyes open under water

You can't see clearly underwater for a couple of reasons. One is the thickness of your lens, but the main one is the index of refraction of your cornea.

For reference, here's the Wikipedia picture of a human eye.

According to Wikipedia, two-thirds of the refractive power of your eye is in your cornea, and the cornea's refractive index is about 1.376. The refractive index of water (according to Google) is 1.33. In water, your cornea bends light as much as a lens in air whose refractive index is

$$\frac{1.376-1.33}{1.33} + 1 = 1.034$$

That means you're losing about 90% of your cornea's refractive power, or 60% of your total refractive power, when you enter the water.

The question becomes whether your lens can compensate for that.

I didn't find a direct quote on how much you can change the focal distance of your lens, but we can estimate that your cornea is doing essentially nothing, and ask whether your lens ought to be able to do all the focusing itself.

For a spherical lens with index of refraction $n$ sitting in a medium with index of refraction $n_0$, the effective focal length is

$$f = \frac{nD}{4(n-n_0)}$$

The refractive index of your vitreous humor is about 1.33 (like water), and the refractive index of your lens, according to Wikipedia, varies between 1.386 and 1.406. Let's take 1.40 as an average. Then, plugging in the numbers, the effective focal distance of a spherical eye lens would be five times its diameter.

The Wikipedia picture of a human eye makes this look reasonable - a spherical lens might be able to do all the focusing a human eye needs, even without the cornea.

The problem is that your eye's lens isn't spherical. From the same Wikipedia article

In many aquatic vertebrates, the lens is considerably thicker, almost spherical, to increase the refraction of light. This difference compensates for the smaller angle of refraction between the eye's cornea and the watery medium, as they have similar refractive indices. [2] Even among terrestrial animals, however, the lens of primates such as humans is unusually flat.[3]

So, the reason you can't see well underwater is that your eye lens is too flat.

If you wear goggles, the light is refracted much more as it enters the cornea - the same amount as normal. If you want to wear some sort of corrective lenses directly on your eye like contact lenses, they should have a refractive index as low as possible.

Googling for "underwater contact lens", I found an article about contact lenses made with a layer of air, allowing divers to see sharply underwater.

Am I right ?

Yes.

If so, what lenses should one wear in order to see clearly while under water ?

You don't need extra lens you have one in your eyes, just use goggles that makes a layer of air between the water and your eyes.

If you decide to put a convergent lens in front of your eyes it won't work because your eye will still not be able to focus different depths. So, you will only be able to focus rays coming from a fixed distance, what is not very useful.

Snell’s law works with a change in the index of refraction across two media. The difference is greater for air on cornea than it is for water on cornea.