I prefer the first. The second is, as you said, awkward.
The third is incorrect.
Interested in Language
1. "Shiatsu" is a Japanese massage therapy that uses pressure on specific acupuncture points to activate the body's energy system.
2. "Shiatsu" is a Japanese massage therapy using pressure on ...
3. "Shiatsu" is a Japanese massage therapy applied using pressure on ...
In the above three sentences, which ones are correct?
I think 2 is incorrect (or at least awkward).
No. ...massage performed by applying pressure ...
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
But I take it that you meant Sentence 4 is incorrect because the verb that modifies "massage" should be "perform", not "apply".
Please tell me if I'm wrong.
I would use:
1. Shiatsu is a Japanese massage therapy which uses pressure on ...
2 and 3 are incorrect.
I would not use number 4 at all. I cannot find a way of making "... massage therapy, applied ..." into a grammatical English sentence.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.