The only way you could list the $5,000 in "other income" as opposed to Schedule C is if it's considered a hobby. This is a common question so the IRS has information on it here. I think the default assumption is that it's a business. In your example, is the student actively seeking out gigs? Then it's almost certainly a business. Or were they just playing for fun and somebody offered them $5,000 to do a show? Then you could potentially make the case that it's a hobby.

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I will issue a caution here: The text above is from the 2019 version of Pub 525, and it seems to be out of date when it discusses expenses related to this activity: With the recent tax reform, hobby expenses are no longer deductible on the Schedule A Itemized Deductions. The IRS page that is linked to in Craig W's answer is dated 2017 and is also sadly out-of-date in this regard.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:

• IRS Publication 535 (Business Expenses), "Not For Profit Activities"
• Title 26 (Tax Code), Section 1.183-2, "Activity not engaged in for profit defined"

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.

It's also worth noting that in both the Tax Code, and in Publication 525, this is not referred to solely as "hobby income," but "Activity not for profit," and the word "hobby" is only used as an example of this. That is why you don't see the word "hobby" everywhere you look. You would need to look at the activity you did which generated the income and decide if it is a for-profit business or an activity that you did without the intent of making a profit.

Here's an example: This week, I helped a friend record and edit a presentation, and he gave me a small gift card as a thank you. Technically, I am supposed to include this on next year's taxes. I'm not going to, because I am not an idiot, but if I did, I would put it on the "Other Income" line. Video recording and editing is not my business, just a hobby; I spend more on tools than I have ever made, I can count on one hand the times I have ever received any compensation for it. I don't treat it as a business; I don't keep books, I don't bill anyone for anything. I wouldn't meet any of the IRS's nine criteria for a business. It's not a business, it is just an odd job (if you even want to call it a job).

For the student in your example, if he spends money on musical instruments and does that recreationally, he does not suddenly have a business just because someone paid him once. It's just "Other Income." However, if he starts to meet some of that criteria, perhaps he starts realizing a profit in multiple years and depends on that income for a living, the IRS would characterize it as a business. It would probably be in his best interest to characterize it as a business at that point anyway, because he would want to be able to deduct his expenses.