There is no real difference between "baggage" and "luggage." They are synonyms. There might be subtle differences, but they are so subtle that native speakers would argue about it. In the United States, airport signs always use the term "baggage." On a trip, the individual containers you carry clothing and personal items in might be called "suitcases" or "bags." "Baggage" and "luggage" are uncountable nouns for a collection of bags and suitcases. "I have to pick up my baggage at the baggage claim area." "How many pieces of baggage do you have?" "Three, two suitcases and a duffel bag." I am not sure what you mean by "things you take to the salon." In US culture, women carry things in small "purses,"and bigger "handbags."

Baggage and luggage are the same, luggage is more commonly used in British English from what I've heard. Some things you bring to the salon: your purse in a handbag or bag and your glasses or spectacles if you need them so you can admire your new hairstyle. Hope this helps! :)

Onboard you can take hand luggage / hand baggage / carry-on luggage / carry-on baggage. You can choose any of these phrases, they mean the same. There's no difference between "luggage" and "baggage" though "luggage" seems to be a bit more common in British English. Oh, also luggage might refer to empty bags with no contents.