a very narrow understanding

From OnestopEnglish.

http://www.onestopenglish.com/Grammar/Reference/Modals/modals1.htm

"Student: Can I go to the bathroom sir?

Teacher: I’m sure you can, but you may not.

This old joke plays on different interpretations of can. The student is asking for permission, whereas the teacher is talking about ability. Some people might say that the student’s use of can is incorrect (i.e. the student should say May I go to the bathroom?). This view, however, is based on of modality and modal verbs in English."

Is the part in blue true?

Sorry, it was meant say:

This view, however, is based on a very narrow understandingof modality and modal verbs in English."

"May" is in some instances a more polite/reserved form of "can" but to insist that "can" only refers to ability is silly and not consistent with how people speak. "Can I go to the bathroom?" or "May I go to the bathroom?" both convey the same meaning, except "may" sounds just a little bit more formal and polite.

I'd agree, they both mean the same thing, with 'may' sounding more formal/polite.

Teacher: I’m sure you can, but you may not.

This old joke plays on different interpretations of can. The student is asking for permission, whereas the teacher is talking about ability. Some people might say that the student’s use of can is incorrect (i.e. the student should say May I go to the bathroom?). This view, however, is based on of modality and modal verbs in English."
ă stands for 'eh-ah, not ah "cat, map, camp, sad..."
ãr stands for 'eh-ah(-r) "care, fair, bear, where..."

ā stands for 'eh-ee "cake, pain, vein, prey, day, sleigh..."
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ĕ stands for eh "bed, sweat, dead, death, health..."
er stands for eh~(-r) "bird, verb, fur, work, word..."

ē stands for ee / e "meat, beet, bee, sea, piece, field..."
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ĭ stands for 'ee-eh / ih "fish, busy, building, bitch..."
ĭ stands for 'ee-ah(-r) "beer, fear, weird, mere..."