"I try to paint the beak, but in the end it looks like a flat, clumsy triangle glued onto the bird's face."
Nearly all bird portraits make a mistake when dealing with the beak, usually for the same reason. Those who are just starting out see a "yellow beak" or a "black beak" and paint it as a single flat area. However, a beak is not merely one colour; it is a three-dimensional cone or cylinder. The light falls on the ridge, the central part displays the main colour and the shadow appears underneath. Only when you paint these three sections do you move beyond simply painting a beak and start to paint a solid object in space.
The good news is that all you need are three values, as long as they are used in the correct order.
When you mix a color, first determine three beak zones: the highlight on the upper ridge, the mid-tone over the main part of the beak (the actual local color), and the shadow where the lower mandible meets the chin.
Begin with the mid-tone and proceed outward. While the paper is still damp, add shadow below it so it blends into the mid-tone. For the highlight on the top ridge, leave a thin strip of bare white paper from the beginning. This unaltered paper gives the beak a hard, shiny, three-dimensional appearance that no subsequent brushwork can match.
A beak that looks stuck on is almost certainly due to an edge problem. Since a beak grows from the skull and is surrounded by small, soft feathers that blur the boundary between keratin and skin, mimicking that transition will make the beak look as if it belongs there.
Begin with the mid-tone, then add the shadow when the paint is still damp and keep the highlight from the start. It is precisely by making those three decisions before you begin that a flat triangle becomes a beak which looks good enough to touch.
Continue learning with the next step in this watercolor series.