grammar - How is あっての used to define something?

I've previously asked about あっての, but now it seems to me that I've run into an additional usage of it (or perhaps the same usage I just can't wrap my head around it...).

When the sentence is simply AあってのB, it seems pretty straightforward. For example:

あなたあっての私なんです。

I think means: "I wouldn't be here without you" This sentence, however is more confusing to me:

子供あっての我が家である。

Does this mean:

• "Our family wouldn't be here without the children/if not for the children"?

Or is it more like:

• "Our family exists for the (benefit of the) children"?

Or even:

• "If we never had children, we wouldn't be a family"?

Answer

I sometimes have this problem with あっての　too so I looked it up (Reference: 日本語表現文型辞典　p25). To my surprise the English explanation was contradictory but the Japanese explanation works.

I'll start with the English definition:

「N１あってのN２」is an　emphatic expression meaning N1 is realized because there is N2.

This fits

「子供あっての我が家である。」

which I take to mean "We would not be the family we are without our children"(see note 3), but seems contra to the more familiar:

「あなたあっての私なんです。 / "I wouldn't be here without you"

However the Japanese explanation uses the expression:

「N１があるからN２が成立する」

which works for both examples.

Notes

1. 
   

   FWIW: The book also give alternative English equivalents as "comprised of" or "indispensable to".

   
   
   


2. 
   

   The full Japanese definition was:

   
   

   「N１あってのN２」の形で、「N１があるからN２が成立する」と強調するときの表現。

   
   


3. 
   

   This translation is closest to your first meaning, which also seems quite acceptable. The others might also work in the right context?