"I read that the withdrawn earnings are both tax- and penalty-free as long as you use them to buy, build, or rebuild a home."

Nope, but it not "up to $10,000" either. That is a life-time maximum, but not the only reason there might not be penalty or income tax.

There are ordering rules: https://www.investopedia.com/retirement/tax-treatment-roth-ira-distributions/

"distributions of Roth IRA assets from regular participant contributions and nontaxable conversions can be taken at any time, tax-free and penalty-free. However, distributions on taxable conversion amounts may be subject to the 10% early distribution penalty. Distributions of earnings that are part of a non-qualified distribution are taxable and may be subject to an additional 10% early-distribution penalty."

"and used the total distribution for purchase of a first home."

"Even if you are under age 59½, you don't have to pay the 10% additional tax on up to $10,000 of distributions you receive to buy, build, or rebuild a first home. To qualify for treatment as a first-time homebuyer distribution, the distribution must meet all the following requirements.