You are right about テクニック being specific.
Another thing to note is 方法 is a generic how to; the others imply some sort of skill one needs to learn. In this sense, テクニック/手法/技法 are quite similar.
技法 and 手法 are inter-defined, so practically synonymous. They mean techniques in art, which are mostly done manually, thus are interchangeable (are there crafts not by hand in some way?). Perhaps 技法 has a connotation of more established/technically distinct collection of methods/skills.
テクニック can be used for skills outside art. For example, speaking of sport player's techniques, テクニック is fine but 技法/手法 would be odd.