In the Sitecore SXA Promo component I want to add a datasource item which is created from another template (custom, not the default Promo template). I want to show fields of that template and use a rendering variant accordingly.
The best thing to do in such a case is to clone the promo rendering (this way you can name it how you want as well): https://doc.sitecore.com/developers/sxa/18/sitecore-experience-accelerator/en/copy-and-customize-a-rendering.html
In the clone, you can use any template as datasource (just change the settings in the rendering).
You probably will want to create a variant for your new rendering as well. In the rendering variants, create a new folder (Variant) that matches your components name exactly. In that folder you can create your variant: https://doc.sitecore.com/developers/sxa/18/sitecore-experience-accelerator/en/create-a-rendering-variant.html