How to add a plugin widget to a sidebar during plugin activation

I found out tons of tutorials how to this, but honestly there is almost bullshit from old wordpress versions and lots of bullshit. So I try my best here to get a simple solution (The right way).

I am also confused about the sidebars. It looks for me that there is no stndard sidebar. Every theme developer names his sidebar in a diffrent way so how can I get sure that my widget is display if the sidebar has a different name? Do I have to build my own sidebar and add there my widget? Lots of questions I know. I am currently a drupal programmer and I miss a little bit the logic in wordpress. So I am open to learn and understand the WordPress way to programmatically add my widget.