What is the Buddhist interpretation of the flow state?

samādhi, undistractible-lucidity, having singular preoccupation, a singular focus would be the closest to 'flow state'. What makes samma samadhi or samadhi-sambojjhanga really start to stand out, is when the passsadhi (pacification) awakening factor is properly done, so one's energy is smooth, relaxed, effortless, which gives samadhi its distinctive quality of excellence in anything it is directed towards. In other words, without passadhi, one could still have samadhi, but with strained, unrelaxed effort the result is of much lower quality. For example, picture a clumsy unathletic person running, compared to an olympic kenyan marathon runner with effortless, relaxed, pacified and perfectly efficient movement.

The state of meditative absorption is called samadhi.

Zen master Sheng Yen distinguished three kinds of samadhi:

• Worldly samadhi can happen during usual worldly activities, e.g. singing.
• Samadhi of Dharma practice, leading to awakening and liberation.
• Mahayana samadhi, which means awakening and liberation applied to helping sentient beings in usual daily life.

It's important to not get attached to samadhi. Even if it feels pleasant, it has transient nature and can become a trap, a dead end in practice.

Sometimes it's called "the trap of solitary peace", or "Mara's cave".

Sheng Yen said that perhaps the best way of Zen meditation (according to the Silent Illumination method) is to not become consumed by meditation but to stay aware.

Our actual way though depends on our abilities and conditions:

One kind attains arahatship through samadhi, and the other attains this state without experiencing samadhi.

As a Mahayana practitioner, I try to develop samadhi — to be more stable under various influences and conditions; but also I try not to attach to samadhi and keep the practice natural, not losing active karuna attitude.

"Being in a flow" doesn't seem to be a proper goal, because I imagine liberation not as being caried by a river; rather it's like hanging in the air, being attentive and open and easily able to move instantly in any direction.

Mindfulness does not mean being aware.

It means bringing & keeping Buddhist principles in the mind.

It appears the question is about fake Buddhism.

Although meditation/immersion is, in general, unification of the mind, one must be careful.

“Yā kho, āvuso visākha, cittassa ekaggatā ayaṁ samādhi;

“Unification of the mind is immersion. MN44

One must be careful here because there is Right Immersion and Wrong Immersion. Flow unifies the mind with a feeling of effortless, peaceful, and expansive limitlessness in the pursuit of an activity. Yet flow always ends and is therefore unsatisfactory. One can actually crave flow--let's go surfing (or whatever) for the rest of our lives. So flow can lead to suffering. A flowing surfer who cannot surf due to old age and illness is suffering. Although flow is peaceful, it is incomplete. We need to look beyond flow.

If we observe any well-practiced Buddhist monk, we see something quite interesting. The robes of a monk flow, but the monk is invariably steady, imperturbable, radiant, and equanimous. The world flows around the monk without clinging.

What is that monk doing?

That mendicant feels inspired by the meaning and the teaching in that Dhamma, no matter how the Teacher or a respected spiritual companion teaches it. Feeling inspired, joy springs up. Being joyful, rapture springs up. When the mind is full of rapture, the body becomes tranquil. When the body is tranquil, one feels bliss. And when blissful, the mind becomes immersed in samādhi. AN5.26

Although this sounds "flowy", notice that the focus is the Dhamma, not some random activity.

Read the suttas, study the Dhamma, learn from good teachers and good friends. May that flow lead to peace.