The scope will say (if not on screen, then in the manual) what the measurement is defined as.
Traditionally it's the time taken to go from 10-90% of base to top. 20-80% is also common.
For certain waveforms, a 63% measurement, or any other definition, can give similar, even identical results, but there is no general case where such a measurement can still apply to any other waveform.
In short, if you aren't measuring it the same, don't expect the same results!
Consider this analogy: suppose you need measure the arc length of a circular segment. Consider three methods:
1. You use a straight ruler to measure the distance from 'start' to 'finish'. Obviously, this works okay for very small angle arcs, but fails dramatically for large angles. (If the total angle is 359.99.. degrees, the measurement is ~zero!)
2. You use a straight ruler to measure many very small segments. You probably want to mark off which points you're measuring between, so you can measure between pairs of points in a consistent manner. But you'll get a more accurate measurement -- if still a piecewise approximation. (You also get a ton of error from lining up each set of marks, but nevermind that for now.)
3. You use a tape measure and stretch it along the arc.
Now consider if, instead of a circular path, you need to measure an arbitrary squiggle path. Method #1 is right out (what if the curve pinches in on itself? It could be off by 10x or more!). Method #2 gives different measurements for different numbers of segments -- until you use so many segments that the curve-per-segment is usefully small, anyway. But that could take a great many points! #3 is the only reasonable one (but, you can only measure one side of the curve -- if it's a piece of metal that's been bent into a shape, you have to measure one side or the other, or both, and neither measurement may be quite the length of the original piece of metal!).
The scope uses method #2, measuring rise time with thresholds placed at particular levels. If you use a different threshold, expect different results, depending on how the waveform goes between those points.

To generalize the concept, you might even ask yourself: under what conditions is -- what we would call a smooth rising edge -- even reasonable? How messy a waveform can you imagine? What if it has a squiggly or ringing "edge", or it's some lumpy thing that doesn't really rise at all? Does a sine wave have a rise time -- should it? It is even useful to speak of? (No, not usually... but maybe.)
Tim