The distinction is about length and use rather than thickness. A jacket is shorter than a coat. A jacket only goes down as far as the waist or hips, whereas a coat covers more of the lower body. A coat is an outdoor garment that is generally longer than a jacket. Coat = Isn't worn indoors. Jacket = Doesn't go down below hip level. As you can see, these aren't mutually exclusive definitions or exact opposites. So, whereas a warmer jacket might be considered to be a kind of coat, a full-length coat never be called a jacket. A jacket can be for indoor wear (such as a suit jacket for business use) or outdoor wear (such as a puffer jacket). A coat is always for outdoor wear. 'Thin' and 'thick' are just adjectives. They aren't part of garment definition. The only time you would say 'Put on your thin coat' is if the other person had two coats, one thick and one thin, and the speaker was using these adjectives to distinguish between them. Or you say 'Wear a thick coat', you are telling the person not to wear a thin coat. 'Thick coat' isn't a type of coat - it's a description that distinguishes it from a thin one.

A jacket is thinner than a coat so if it was not too cold out you would say " Put on your jacket". If it was very cold out you would say "Put on your coat". We would not really comment on thin/thickness unless-"It is too cold out for a thin jacket, you should put on your coat".

Not exactly. A jacket is usually referred to a light coat. A coat will be thicker.