Maybe we do need another word besides “rigor”, but “challenging” and “rich” are weak alternatives.
Rigor in my teaching practice means conscientious excellence. For example, rigor requires students to differentiate solution from answer. Suppose the question is, “What are the best dimensions for a particular garden?” The student will algebraically calculate two perfectly legitimate solutions. One solution will show two negative numbers; the other will show two positive numbers. Students must choose the solution which answers the question. Since the question is about a garden, that would be the solution with the two positive numbers,because a garden cannot have negative measurements. A different question might require the negative solutions to be the answer.
Or perhaps the question is, How many cars do we need for the field trip. The solution might be 7.2, and it can be the correct solution, but the wrong answer. The correct answer is, "We need 8 cars." Mindless rounding also yields a correct solution,but a wrong answer. I require answers written in complete sentences that also include the unit. I would mark all three of these so-called answers wrong: "x=8", "8," and "We need 8."
I also require students to keep units attached to numbers when they calculate. So the area of a room is not 9X12, with the ft^2 attached later. When students show their work, I want to see 9 ft x 12 ft = 9 x 12 x ft x ft = 108 ft^2. Please do not dismiss my simple example as trivial. This sort of training, call it rigor if you like, pays off big when students must do chemistry or physics calculations with lots of units running around. My physics students learned that if the resulting unit is not what they expected, they probably also made a more serious mathematical error somewhere. That self check is lost when units are divorced from numbers and remarried at the end of the calculation.
In the earliest grades, rigor may imply making sure students understand the role of an equal sign, and knowing that the horizontal line separating a column of numbers from the result of a calculation is not a substitute equal sign. Many adults graduate from high without a proper appreciation of an equal sign.
Every field has similar examples of the value of conscientious excellence. Most people prefer the two syllables of “rigor” over the seven syllables of “conscientious excellence.” Just because three of the definitions seem negative and harsh does not invalidate the value of the fourth definition. Rigor, properly used, is not a blockade to academic achievement or educational accessibility, but its open door.