David has it right. Take it from someone who taught the stuff for longer than he would care to remember. Even when considering tuning schemes other than equal temperament you'll generally find the terms used this way. However, please note that while the numbering of half steps on the chart is correct in describing the size of the mentioned intervals it can be confusing. I think that is what Geoff is referring to. David's example is a good one. If you think about the letters the number of letters involved in the interval (counting upward) determines unison, second, third, fourth etc. So, any C (Cb C# C or even Cbb or C##) to any E (I don't have to repeat the possibilities do I ? ) is some kind of third. What kind can be determined by counting the half steps and following the conventions so C-E is Major 3rd, C-Eb minor third, C-Ebb (or C#-Eb) is a diminished third, and Cb-E (or C-E#) is an augmented third. You can do the same thing with other intervals too. Things can get confusing though. Consider F-B is an augmented 4th while F-Cb is a dimished 5th. Or maybe don't consider that
I hope this clarifies a bit and doesn't just add to the confusion. Please note this does not consider tuning issues which, as pointed out above, can result in different sizes of particular intervals so that D#-E and Eb-E may not just have different names (minor second and augmented unison), but also slightly different sizes when measured in 100ths of a semitone (cents). That is a whole different issue.