history -cw does not overwrite ~/.bash_history file

I use Arch Linux and bash. I've identified the following and can't understand how it works. The first issue is: history -cw doesn't clear ~/.bash_history file but history -c ; history -w does. The second one is: When I open the shell and execute something like these:

$ first
$ second
$ history -c
$ third

and close the shell, ~/.bash_history is supplemented with a line third. I expect ~/.bash_history to contain the single line third. My shopt histappend is off, doesn't this mean that the history file needs to be overwritten? It's not critical for me, but I want to understand how it works.