SemEval-2024 Task 8 is focused on multigenerator, multidomain, and multilingual black-box machine-generated text detection. Such a detection is important for preventing a potential misuse of large language models (LLMs), the newest of which are very capable in generating multilingual human-like texts. We have coped with this task in multiple ways, utilizing language identification and parameter-efficient fine-tuning of smaller LLMs for text classification. We have further used the per-language classification-threshold calibration to uniquely combine fine-tuned models predictions with statistical detection metrics to improve generalization of the system detection performance. Our submitted method achieved competitive results, ranking at the fourth place, just under 1 percentage point behind the winner.