No, there is a specific order in which Roth IRA distributions must be taken. 

Contributions to the account are distributed first.  Therefore you cannot decide that your distribution is from the earnings in the account instead even if you do have an exception to the early distribution penalty.  

Remember also that even though a withdrawal of earnings would be exempt from the 10% penalty for a first time home buyer, you would still pay regular income tax on any withdrawal of earnings that is before the age of 59 1/2.  The withdrawal of contributions is tax free and penalty free at any time for any reason.

Since the $10k distribution would be coming entirely from contribution basis, there is no need to claim this and no benefit to claiming this as a first-time homebuyer distribution.