You have answered your own question. A second home would not qualify as a first time home. I am not aware of any tax credits for any first time home buyers although I do believe that there are some mortgage products for first time buyers.
$10k from a 401(k) fund may be used for a first time home purchase without incurring the 10% excise tax but remains taxable as current income. The definition of first time home buyer is fluid as it means any person who has not owned a home for at least 2 years.
Either way... I don't see how any of these would apply to you.
Warmest regards John
Edwin, you may be referring to "exclusion on gain of sale". This is a feature of the tax code that allows many homeowners to recognize a gain on the sale of their home, without having to pay any tax on that gain. Most homeowners will be able to make a profit of as much as $250,000 on the sale of their home ($500,000 if married filing jointly). However, this would apply only to your PRIMARY residence- not on a second home. You still can deduct mortgage interest on that second home and that may still be a help at tax time. For instance, if you didn't have sufficient deductions to be able to itemize before, the interest on the second mortgage would probably do it. Before finalizing your plans you should consult someone well versed in tax matters.