I understand where you're coming from: all the info about TEFL can be very difficult to digest at the beginning. Believe me, you are far more focused than many people who are starting out. You have already decided which place you would like to work in and have narrowed down your choice to 2 groups, teenagers and children.
The fact that you are focused on one place and 2 age groups will make things easier: you can start looking at job ads in Hong Kong and see which qualifications are mentioned. However, you should treat what you find there as a guide not as a definite rule when choosing your course. Remember that the course will serve you for the rest of your TEFL career, you might not want to work in Hong Kong for ever.
The two courses that are most-widely known are: the CELTA and the Trinity certificate. The fact that they are the best known means that employers recognise immediately what they are dealing with. This obviously has advantages when applying for work but it doesn't mean that they are the only valid qualifications. As for accredited organisations, you can check with the British Council.
As for the difference between TEFL and CELTA, that's very easy. One is a qualification and the other is a subject. TEFL means teaching English as a foreign language. It is a subject that teachers study or an area that teachers work in. CELTA is the certificate in English Language Teaching for Adults. This is a qualification.
Please write in again if you have any other questions.