If I have not applied for a passport in some ten years, should I "re-apply" or "apply for the first time"?

They know that you've previously held a passport so you wouldn't pretend it's your first time. You're no longer a virgin ;-)

As you already have had a passport it is obvious you are renewing, as it is certainly not a 'first' passport

You are NOT applying for the first time, you are already on the system even though it was years ago.

So you renew. Your passport has a number, they renew it.

UK

You need to apply as if it is your first time. 

Assuming you are asking about a US passport then you are applying to renew your passport. Despite the use of the word "renew" you get a new passport with your current photo.  

https://travel.state.gov/content/travel/en/passpor...

If you do not have a currently valid passport to renew, you start from scratch as if applying for the first time.

You would be reaplying.

The point is have you had a passport previously, you have.  They didn't ask when is the last time you've had one - they only asked if you have had one.

Are you applying for the first time - obviously not, if you have applied for one before.

If you apply for a new passport after you had one years ago, it might look like you're trying to hide something.  I'm pretty sure the Fed. Govt. has a record of your passport.  They don't forget after 10 years.  I would reapply.  If you then got a letter saying "Why are you reapplying?  We have no record of your passport before"  THEN you apply for a new one.