The Special Edition PHB disambiguates the wording. Some people refer to this as "stealth errata" since it doesn't seem to have made it into any other sources, nor was it conspicuously announced anywhere. Even the SRD doesn't seem to have been updated to contain the new wording. Nevertheless, it makes the intent of the feat clear. Emphasis mine:
When using the attack action with a melee weapon, you can split your move action
in that round in order to move both before and after the attack, provided that
your total distance moved is not greater than your speed. Moving in this way does not
provoke an attack of opportunity from the defender you attack, though it might provoke
attacks of opportunity from other creatures, if appropriate.
You can't use this feat if you are wearing heavy armor.
You must move at least 5 feet both before and after you make your attack
in order to utilize the benefits of Spring Attack.
Notably, this is still not the same as Pathfinder's implementation: instead of combining the attack and move into a single full-round action, it keeps them as two distinct actions but allows you to apply them in an unusual way (much like the Overrun writeup that you cite). But neither implementation allows you to make a spring attack using a single standard action.
If you want to houserule Spring Attack to take a single standard action, go right ahead. No one will stop you. I might consider such a thing myself in my game, if it ever comes up. But that's not RAW, and in one of those rare cases where RAI is demonstrable, it's not RAI.
Although movement in 5e is not governed by squares, they are probably a helpful model to this particular exercise. Everything in the universe is effectively measured in 5' increments and using squares to model this will give us an effective answer to this question Let's posit a 50' wall
x
x
x
x
x
x
x
x
x
x
Now let's look at the veritcal space next to the wall
yx
yx
yx
yx
yx
yx
yx
yx
yx
yx
Finally we need to have the flat part on top of the wall, and the space above that
yyyyyyyyy
yxxxxxxxxx
yx
yx
yx
yx
yx
yx
yx
yx
yx
Now, supposing our hero starts at the first y (marked a below), and runs up, he runs out of movement at the 10th y (marked b below).
yyyyyyyyy
bxxxxxxxxx
yx
yx
yx
yx
yx
yx
yx
yx
ax
If you start at a, you might be able to climb up onto the top of the wall. However, if you have a 45 foot wall you are safe. If you have to move to a, a dash would be required to not fall (as you don't have the ability to stay vertical).
The problem here is that if the wall extends at all into the square above the one marked b, then you're in trouble, you have to move into the square above it as diagonal movement is not possible.
This will be both at the discretion of the map makers (if the building extends a bit above the square you're probably not going to move diagonally) and also your DM (he may allow a bit of wiggle room here). Consult your DM before you attempt any 50' wall climbs.
45' wall climbs (and lower) are safe though.
Best Answer
Spring Attack is not a charge or usable in conjunction with a charge. Spring Attack describes a unique full-round action, while running or charging are both their own full-round actions. You cannot combine actions together; if you are using one, you are by definition not using the other at the same time.
You could modify water run to include Spring Attack, perhaps like so:
You could go further by adding straight-line requirements on the usage of Spring Attack, too:
At which point it might be simpler to just say
Which would include running and charging by definition (since each of those require moving in a straight line), but also any other kind of movement where the creature moves in a straight line. Note it does have the drawback of no longer allowing turns even when a creature would otherwise be allowed to turn during a charge (e.g. using the Twisted Charge skill trick); further rules could be added to water run to allow those, though:
As for official abilities that allow moving after a charge, the only one I know of is the Ride-by Attack feat, but as the name suggests that is only while mounted. It would not be compatible with the water run ability (since the mount wouldn’t have it).
Giving a race an unmounted Ride-by Attack ability would be a pretty good benefit, but hardly unreasonable.
You could also try things like allowing them to count as mounted for feats in general, and then as having Mounted Combat and Ride-by Attack for prerequisites. Something like this:
Might have gone overboard on this by this point, but just to give ideas. Feel free to pare this down to a simpler feature as your tastes dictate. Removing Ride-by Attack as a bonus feat would also greatly reduce the power, maybe make it more appropriate with water run. Then the race wouldn't have the ability by default, but would have the option to take a feat for it.
Bear in mind, though, that water run’s significance will drop off quickly, and by mid levels (5-10-ish) it will probably be pointless: by then, most every character should have a way to fly. At low levels, it’s pretty potent, and water run plus powerful run might seem too good. But if you nerf things too much, by mid levels the race may seem rather lackluster.