After you file

Can you direct to IRS sources to back up #4 household employee-regarding how much a full time student under 18 can earn babysitting (lawn mowing etc.) under the under 18/Household Employee rules? 

 

I am getting told that my kid needs to claim as self employment if above $400, and thus pay self employment tax. But I don't think that is right--because, if household employees/w2 are excluded from paying SS/Medicare IF under 18/full time students why should self employed (non w2/non1099) teenagers under 18 be paying?  I can find the rules that apply to W2 household employees under 18/students, but not those that are typical teen babysitters under $600 per family, multiple families.  

Also, if this is typical date night babysitting and not a "nanny" job --is it still supposed to be entered under Misc. Income non W2 Household?  Or, where should it be entered?