Job done, the only thing needed was determination.
Switched my hosting provider to one with better support and platform.
Updated my domain dns towards my new hosting and also requested domain migration.
Checked some reviews about blogging software and picked WordPress. Installed a theme and some plugins.
Modified the theme a bit, took longer than expected to configure the teme, pages, links, posts, body, search…, not very friendly to be honest.
I guess less experienced users would have taken longer than me.
When everything was finished, I apparently edited something wrongly and nothing worked anymore. After reading a bit, I found that the editor automatically left some spaces before the php tag… crazy editors of platforms that run horrible languages turn people nuts.
And… voilà! I just need to spend 30 minutes every time I want to share something with you.