If I hit F4 in a folder, that folder is opened in terminology.
If I do the same as sudo, there's another terminal with a too small font size.

How can I fix this in order to have the ordinary terminology there, too, please?

Sounds like you have a different default terminal emulator set for each user.

Try this as root

sudo update-alternatives --config x-terminal-emulator

Should allow you to choose the default terminal emulator

I didn't do anything about the terminal, it's just as it was when I installed this system.

Hmm maybe not intentionally but there is no other terminal installed on BL 5.0 other than terminology so if another one opens such as xterm then you have installed it. Maybe it was brought in as a dependency or recommended package of another app but nonetheless it has been installed intentionally or not. And it is usually the case that installing a terminal will run update-alternatives in the post install script. If you pay attention to things you will see that in the terminal output when you do the install. It is also apparent if you look at the deb file itself.

In fact I added such a command to the post install script of our terminology app because I got tired of seeing x-terminal-emulator not set by default in the BL isos Jeff made. This meant open in a terminal was failing to work in pcmanfm until one either changed pcmanfms settings or one set x-terminal-emulator.

Anyways just an explanation ...

Doesn't Bodhi use pcmanfm? You can set the terminal in the preferences.

Yes pcmanfm is the file manager installed by default and yes there is an option to set the terminal in pcmanfm preferences. By default pcmanfm has it set to x-terminal-emulator but of course one can change that. Changing options running a gui program as root can mess up your permissions in your home folder, one of the many reasons people say don't run an app using the command sudo appname.