ANSWER REVISED
No, it's not true. For a household employee in 2022, the employer had to withhold Social Security and Medicare taxes if the employee earned $2,400 or more. That might be where the $2,400 figure comes from. (The amount is $2,600 for 2023 and $2,700 for 2024.)
For any other employees, there is no fixed income level at which withholding starts. It would depend on your filing status, number of dependents, other income, deductions, etc. You fill out a Form W-4 and give it to your employer. The employer uses the information from the W-4 with some tables or formulas to determine how much tax, if any, to withhold. Social Security and Medicare taxes have to be withheld, even if no federal income tax is withheld.
