You are correct that Baby in the child sense is a newborn to 2-3 years old or so. Probably more 2 than 3. It isn't limited to someone's own baby at all - it is a general term. It can also be a term of affection (e.g. to a girlfriend), or as a derogatory term, to show that someone is childish. A child is usually from a baby to a teenager - the only difference between child and children is that one is plural. If you are using it personally 'I have one child' - it can be any age at all - even up to 60 or more. Kid/kids is just an abbreviation/slang term for child/children, so it refers to the same age group. Yes, if you were calling out to a group of children in an informal situation, you would usually use 'Kids!'. A teacher, on the other hand, might use 'children!'.

You're close. "Children" is general for the offspring of parents, including when they are adults. "Kids" is informal for children usually when they are less than 18, or perhaps older if they still live at home and are young adults. "Baby" is normally used for children between new-born infants up to when they start walking, at which point they become "toddlers".

Great answers, thank you both!