As far as I know, it wasn't explained in the anime (though I'm not that sure since I haven't finished Mitsuganae and I haven't read the manga). But I think, it works like this.. (this is purely from my own deductions since I haven't found anything clear on any sites I searched).
If a person send someone to hell by pulling off the red string from the straw doll Enma provided, that someone will immediately be ferried to hell, thus completely disappearing along with the person's body. After that, the sender will gain a mark on his/her chest signifying that his/her soul will go to hell after he/she dies.
So, if a person is sent to hell, that person will disappear, along with his/her body, leaving nothing at all in earth. If a person was a sender, once he/she dies, his/her soul will go to hell, but his/her body will be left on earth. A sender will only completely disappear if he/she was sent by someone else to hell before he/she dies. That, I think, is how it works, in most episodes.
In Episode 13 Purgatory Girl, the two persons you mentioned who died were Fukumoto's wife who committed suicide and Okochi, Fukumoto's friend whom he sent to hell. Fukumoto's wife wasn't sent to hell, she committed suicide. Okochi however was sent to hell. The man (I forgot the name) who was retelling their story indeed said "Okochi died." in the english subtitles but what he said in japanese was "Okochi-kun ga nakunatta.." literally translating it means Okochi-kun disappeared. The man narrating the story didn't mention anything about Okochi's body being found or something like that. And it was shown that Fukumoto who sent Okochi to hell died and his body was left in front of his final artwork.
However, Season 2 Episode 19: Steamy Hell seems to be an exception. It indeed showed the burnt body of a person who was sent to hell instead of having him completely disappear.
I don't know why it was like that in Steamy Hell (probably a plot hole or something) and there is no definite explanation or answer to your question. It can be just assumed that what I've explained above is the general rule and Steamy Hell is the exception to that rule. Or, there is just no rule whatsoever for that. It probably depends on Enma or her boss (the spider) on whether they will leave the person's body on earth or not.