Should the term 廁所 (bathroom) be used for both a home and public toilet?
You can definitely say "廁所" even it is in your house. There are more than one way to say it. The word "廁所" and "衛生間" probably are most common ones. Other than that "洗手間" is pretty common too. "盥洗室" is not very common in daily uses. You normally will see this in the hospital or kindergarten. The word "浴室" might not be the best one for bathroom because it focus more on the cleaning part. It usually indicate the place that you take a bath. In some area in China "浴室" normally means public bath place.

Again, different place in China has different language habit but "廁所" is definitely ok.