What do Medium and Hard mean in this context? Do they correspond to DC descriptions of Moderate (DC 15) and Hard (DC 20)?
Here's a simple take:
Character makes an Athletics check.
What you roll is how far you jumped in feet, minimum equal to STR, maximum of STR x 2.
Halve, third or triple the distance as appropriate for standing/vertical jumps and magic.
Keep in mind real world athletes can make almost 30' long jumps.
I might adjudicate things some other factors on the fly, like Disadvantage for heavy Encumbrance or Armor, close rolls mean you grabbed the edge (fail by only 1 or 2), or land prone (succeed by right on or only 1 or 2), etc.
Playing 5.5? Have something comically tragic happen on a Natural 1.
And if I'm feeling mean, Small characters treat STR as -2. If I'm particularly mean, Dwarves are included in that.