It’s not about specific phrasing, it’s about giving the DM enough information to make a judgment call about the results.
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“I make an Athletics check to jump further than I normally can” actually does give me enough information. Your goal is to cover more ground than you can normally jump, and your method is by jumping. From that description of what your character is doing, it is my job as DM to determine what the results are, and skill checks are a tool to help me make that determination in the case of an uncertain outcome. Personally, I would say that your approach, “jump” does not have a reasonable chance of success at achieving your goal, “cover more ground than I can by jumping,”