Originally, a dressing gown was literally something you wore while dressing. You would not want to get hair powder or Macassar oil on your delicate silk or wool garments, so you would cover them with your dressing gown until 'you' finished doing your hair. Or, as dressing could be a lengthy, complicated business, your maid or valet could give you your dressing gown to wear over your incomplete outfit if someone wanted to see you in your boudoir, and you did not want to be seen improperly clothed. It was a light garment that could cover or expose as much as needed.
A housecoat had a similar purpose: it protected your 'good' clothes. I remember my mum, who ran a business from home, would come home from a meeting, take off her smart suit and put her housecoat on over her tights and slip. Then she would get on with the housewifing and mothering, alongside fielding business calls, until she needed to leave for her next meeting. The housecoat would come off, the smart suit go on again, and off she went. DM's ankle-length, quilted housecoat buttoned up to the neck. Nothing showed.
A bathrobe was a thick towelling robe that men wore when walking to the bathroom, which could have been at the end of a corridor, or even in another building. Or an outdoor pool. Ladies were not expected to expose themselves in that way.
Nowadays these distinctions have bcome irrelevant. Our clothes are less complicated, less fragile, and easily cleaned. Our bathrooms are across the landing, or even in our rooms, and our houses are warm.