There are lots of types of income that go on the Schedule 1, Line 8 "Other Income" spot. The Form 1040 instructions say this:
Use line 8 to report any taxable income not reported elsewhere on your return or other schedules. List the type and amount of income. If necessary, include a statement showing the required information. For more details, see Miscellaneous Income in Pub. 525.
It then lists lots of examples, but they are just examples, and it is not an exhaustive list.
Going to the aforementioned Publication 525 (Taxable and Nontaxable Income), in the section named "Miscellaneous Income," under "Other Income, Activity Not For Profit":
Activity not for profit. You must include on your return income from an activity from which you don't expect to make a profit. An example of this type of activity is a hobby or a farm you operate mostly for recreation and pleasure. Enter this income on Schedule 1 (Form 1040 or 1040-SR), line 8. Deductions for expenses related to the activity are limited. They can't total more than the income you report and can be taken only if you itemize deductions on Schedule A (Form 1040 or 1040-SR). See Not-for-Profit Activities in chapter 1 of Pub. 535 for information on whether an activity is considered carried on for a profit.
I will issue a caution here: The text above is from the 2019 version of Pub 525, and it seems to be out of date when it discusses expenses related to this activity: With the recent tax reform, hobby expenses are no longer deductible on the Schedule A Itemized Deductions. The IRS page that is linked to in Craig W's answer is dated 2017 and is also sadly out-of-date in this regard.
There is no official limit or threshold at which you have too much hobby income to include here. The key is to determine whether or not your activity is a "hobby" or a "business," and this is a complicated issue. The general guideline is whether you are intending to make a profit with the activity, or whether you do it solely for fun. The IRS has nine criteria that they use to determine whether or not your income-producing activity is a business.
More Reading:
Addressing the three references in the last section of your question:
For each of these, the fact that they do not reference hobby income in the lists of "examples" does not imply that hobby income does not go here, as these are not exhaustive lists. The Form 1040 instructions for this line that I reference above states, "For more details, see Miscellaneous Income in Pub. 525," and I do just that in the next part of my answer above.
For the article from The Balance that you reference, one of the examples is "Cash earned from odd jobs," and the $5,000 that your example student made in a gig would seem to me to be a part of that.