We use APPARENTLY to say that we think something is probably true because somebody has told us or we have read or heard it.
We use PRESUMABLY to say that we think something is probably true because we have thought about it and used our knowledge and understanding of the world.
- Apparently, more than three quarters of all car accidents are caused by drivers who are either drunk or under the influence of drugs. (= I saw this online, in a newspaper or on a TV documentary.)
- Presumably, when they get into their cars, drivers just do not realise how much the alcohol or drugs have affected them. (= I think this because I have seen drunk people and I know how they often behave.)
- Apparently, most children these days have serious issues with their teeth. Even primary-school children often have to have a tooth taken out. (= I read this in a magazine.)
- Presumably, their parents allow them to have lots of sugary sweets and treats and this causes the problem. (= I believe this because I know that young children like sweets and that sugar causes tooth decay. My experience of the world lets me understand this situation.)