You are mixing world religions and confusing Chinese chi with Japanese ki. This is problematic. They are different interpretations of mystical force.
Since you used the term "kiai" I will speak from the Japanese interpretation since kiai is a Japanese word.
The usage and meaning of kiai varies depending on context and origin. The Japanese have many thoughts on kiai and it is represented mostly in martial arts.
In martial arts you either have physical styles that focus and destroying the opponent or you have spiritual styles that focus on metaphysical analysis of phemenoma that can not be explained readily in western scientific terms.
Physical styles vary in their approach but depending on level of eliteness, ki starts with a physical change in physiological structures that can be measured scientifically i.e. punching harder. Proper biomechanics through breath control allows muscles to tighten in the diaphragm allowing a more rigid body allowing more force to travel through targets. This is similar to specialized breathing done during bench pressing weights or rowing where the breathing action is coordinated with the movement and breath is held so that the body is harder allowing more weight to be moved. This is superficial ki but beneficial to fighters.
Spiritual styles more or less worship ki as a connection to the divine and use meditative measures to establish a connection with the divine. They are trying to tap into that reservoir you referred to. Kiai is used in all Japanese martial arts not just striking based ones like karate, but it is also used outside martial arts and can indeed be found pretty much in all areas of Japanese life, especially those that are greater influenced by Shinto Buddhism.
Karate and jujitsu are heavily influenced by zen and buddhism. In Sanchin kata you are training to be centered and have a connection to the ground. You picture a "ball" in the middle of your body just below the navel and this is attached to the core of the earth thus making you "rooted". This "rootedness" shares likeness with seiza or zazen, there is this idea that when in seated position you are bound to the earth.
From a martial arts perspective mixing highly specialized arts with each other leads to compatibility issues and creates weaker arts. They become too esoteric. It is like mixing world religions and then claiming you have the best religion. It is just too agnostic. Chinese chi or qi is not ki and hara is not dantien. It would be like calling mormons baptists or protestants. Yes there are similar generalities but you would not consider them to be interchangeable terms as they have different pedigrees and take views and practice far apart from each other despite having common roots in a parent system.
What does kiai mean? The meaning of kiai has no formal definition. It's exact meaning is based on individual personal views. You can not "codify" kiai. In Japanese culture learning travels in a circles where when you master something you end at square 1 the beginning, a take on rebirth and starting anew. From not knowing what kiai is you will learn and that experience will shape your definition. When you have mastered what kiai means to you will be at step 1 supposedly ready to take in new knowledge (from the universe).
"Dantian is the 'Sea of Ki'" does not exist, you mean to say Dantian is the 'Sea of qi'. This is not appropriate to switch in an out of Chinese to Japanese as they are separate cultures with their own formalized opinions. It is like speaking "Spanglish". You are botching both systems and lacks harmony and misrepresents each separate style as one. Kung fu is not karate but mixing them and calling it "karate fu" is sure to raise eyebrows since this portmanteau would be seen as comical wordplay and nothing else.
Raising one's ki out of the body refers to Aikido and that is called bugei or "ghost technique". To take aikido and make it practiced in a physical way implies breath power or dragon power but there is no kiai To practice it in a spiritual way is more philosophical and more like a religious cult with meditation and ceremony. Taking this ceremony and habitual ritualistic repetition and combining it with tai chi it for entertainment purposes only. This would be similar to televangelism, you are commercialing the concepts for a broader mass appeal.
Etymological coincidence? No. More like etymological error. The Japanese took their alphabet from China. The two languages have similar sounds and words but vastly different meanings. The two share no connection except a distant past. Research how the Emperor of Japan created the Japanese language and how the Chinese misunderstand each other through speaking Mandarin and Cantonese. These three languages are not transferable in meaning.
Kiai on a basic level is yelling when punching in karate and karate only (maybe judo when throwing). It exists nowhere else (except maybe grunting while swinging a tennis racket or lifting heavy objects). Karate is dense and obtuse and does not go in for Chinese medicine. It is too simple and pragmatic to deal with fancy notions like that. Upper class sophisticated types are more into philosophy and drawing conclusions of the metaphysical with the physical world but do not engage with manual training and practical application of karate. Ki exists in kiai as energy but there is no kiai in ki. Qi is not ki and definitely does not have kiai in it or anything like it. Saying it does is a poor representation of our current understanding of physics.
If you are intent on creating a agnostic one world religion then the kiai is the third chakra, which is associated with sexual organs the loins and intestines, and to be conscious or aware you need insight from shape shifting reptiles would will teach you forbidden knowledge. The legendary Musashi is said to have defeated these creatures. The only battle he ever lost is said to be by the inventor of the jo (aikido weapon) who learned how to make it from the same creatures who come from the abyss (abyss means portal to another dimension but is represented as a whirlpool or other body of water). Physics cannot find these portals. As such demonology and spirit worship comes into play along with themes of lgbt. See wikipedia https://en.wikipedia.org/wiki/Kappa_(folklore).