Could someone tell me the usual terminology for moving an interval up vs moving an interval down vs moving an interval distance? When we write things like 1 b3 5 we mean moving up modulus octaves, right? But when we say an interval is a third, do we mean it is a +3 (like  to ), or a -3 (like  to [b6]) or do we mean "a third" as in a distance measure or magnitude |3| rather than a vector quantity?
Saying that an interval is "a third" only specifies the magnitude, not the direction. However, without context, intervals are presumed to be measured from the lower note. To specify direction, one would say "an ascending third" or "a descending third".
Scale degrees are presumed to be within an octave unless otherwise specified. But direction can be ambiguous. 1 b3 5 is presumed to be a series of monotonically ascending pitches/intervals. 1 b7 b6 would be presumed to be a series of monotonically descending pitches. But 1 6 5, for example, would be ambiguous.
It is important to distinguish between harmonic intervals (simultaneous), and melodic intervals (consecutive).
Harmonic intervals are counted from low to high, so are implicitly upward (unless otherwise specified) — like a magnitude.
The interval between C4 and E4 is a major third. '' '' '' C4 and A4 is a major sixth. '' '' '' A3 and C4 is a minor third.
Melodic intervals are given a direction from the first note to the second, either ascending or descending (up or down) — like a vector.
The interval from C4 to E4 is an ascending major third. '' '' '' E4 to C4 is a descending major third. '' '' '' C4 to A4 is an ascending major sixth. '' '' '' A4 to C4 is a descending major sixth. '' '' '' A3 to C4 is an ascending minor third. '' '' '' C4 to A3 is a descending minor third.
Compound intervals reduce the name of the interval to be within one octave. This is usually more useful; it makes them easier to work with / think about.
The interval between C4 and E5 is a major tenth, or a compound major third. '' '' '' C4 and E6 is a major seventeenth, or a compound major third. etc.
NB intervals are ordinal numbers, not cardinal numbers.
Intervals are always calculated from bottom to top. But, as phoog points out, a 'third' is too vague as it stands. C>E is a third, but so is C>E♭. That's without considering C>Ex, or C>E♭♭. (Major 3, minor 3, augmented 3 and diminished 3 in that order). So 'third' needs to be specified.
It's easy to say 'up a M3', or 'down a M3' to anyone who understands - it's unequivocal: C>E, or C>A♭.
Just be aware that the inverted interval is different - C>E is M3, while E>C is m6. The 'rule' is simple - it's the 'rule of 9'. (3+6=9), M becomes m, aug becomes dim (and vice versa). But any given named note to any other named note will always have the same interval name, totally regardless of key.