The other comments have covered a lot of the background and variances throughout the world. But what I’ll add is that few countries are purely in one camp or the other. To use the USA as an example, criminal cases are adversarial, in the sense that the defense attorney will duke it out with the government’s attorney whether someone goes to prison.

For civil cases like a contract dispute, the procedure is closer to an inquisition system, although with the judge still merely presiding over the process. But attorneys in a USA civil case can depose witnesses, much like how (I think) a European judge-led inquisition would call a witness, and similar to how British coroners conduct an inquest (if murder mystery depictions on the BBC are accurate).

Perhaps the full thrust of the inquisition style can be found in USA federal agencies, whose rulemaking capacity requires asking direct question to subject matter experts in a public forum, one which eventually leads to a determination on some germane topic, often enacting secondary legislation at the same time. Americans might not necessarily call such an action as a “ruling”, but evidence was taken, all sides were heard, and even public comment was accepted, before rendering a decision.