Considering about symbolism, I found the flow of ink quite similar to the concept of thoughts and memory. That video containing red ink starts with a complete blank, which suggests the memory is missing at the beginning or the character is as innocent as a newborn without hard memories. Every drop of ink in the water indicates the gaining of power, thoughts and memory. When the water is filled with ink, that could have several symbolic explanations. It may indicate that the character began to have this illusion of thoughts and incorrect memories. Another explanation could be the character is becoming a person with rich living experiences and also being burdened with memories of himself and others.