Do you wish there's a god?

[Acrobat]

What qualifies as intrinsic purpose? If you define it as only god-related purpose, then of course, no god means no intrinsic purpose. But that’s just a tautology. This is the final kind of nihilism definition I was referring to.

I could argue that even god-purpose isn’t intrinsic to life itself, but in fact it’s our output which has purpose to god, or whatever.

This comes back to the whole problem that purpose is subjective.

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PS: it’s an argument about facts versus value judgements. Facts are objective, value judgements are subjective. To which does nihilism refer? If it’s the former, then it can be trivially easy to demonstrate some fact or other which is supposed to show there is "no purpose". Whether or not the definitions used are convincing enough to carry the weight of the semantics would be up to the reader though. I can similarly demonstrate a fact which I claim shows life does have purpose. It all depends on what someone counts as purpose.

If it’s the latter, then there are no facts, only opinions and subjective evaluations.

No, not all values are subjective. 

"Moral Realism (or Moral Objectivism) is the meta-ethical view (see the section on Ethics) that there exist such things as moral facts and moral values, and that these are objective and independent of our perception of them or our beliefs, feelings or other attitudes towards them. Therefore, moral judgments describe moral facts, which are as certain in their own way as mathematical facts."

If some values, like moral values are objective and independent of our perception of them or our beliefs, than your claim that they're all subjective is false. In fact if you want to go down the line of arguing that there are no objective moral truths, the same exact argument could be used against non-moral objective truths.

As atheist philosopher Louise Anthony put it “Any argument for moral scepticism is going to be based on premisses which are less obvious than the reality of moral values themselves.”