What is the significance of the 这一天 in the longer version?
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"每到教师节这一天，..." refer to the actual day, and the actual day alone
"每到教师节，..." can refer to the actual day or the time period around that day.
If you said: "Every Christmas, all the family members would come home from different parts of the country." It gives room to the possibility of the family members arrive in different days. (may be a day early or a day late)
If you said: " Every Christmas day, all the family members would come home from different parts of the country." then it must mean all of them arrive on Christmas day.
每到教师节, 心意咭的销售量都会比平时大几倍 (the increase of greeting card sale wouldn't occur just on the actual day, it would happen days before.)
In short, "每到教师节这一天" refers to one specific day only; "每到教师节" can be used as a shortened form of "每到教师节(期间)"
There is no difference in meaning. Think of it as the difference between
Admittedly, this isn’t the best analogy, as Christmas can refer to the period of festivities leading up to Christmas Day.
這一天 is the kind of addition you may find in spoken language, as 每到敎師節 is the kind of phrase that has a chance to be misinterpreted due to homophones.