Back linking: Relevance or Domain Authority?

[Modbargains]

For back linking, what is more important for increasing rank for organic searches on Google?

I've noticed that linking on sites with high domain authority that have some or no relevance boost my keyword ranking. However, does back linking on relevant sites help if their domain authority is not so high?

[EGOL]

Normally... if I had a choice between a link on a relevant PR2 page and a link on an irrelevant PR5 page I would opt for the PR2 link.

However, if I had a choice between a relevant PR7 page and a PR2 page on the Pope's new site (which would not be relevant to my site for sure! )... I would take the link from the Pope.

... just sayin' what I would do.... no proof on what would really happen.  I think that the Pope would give a kickass link! 

[Modbargains]

I would take the Pope's irrelevant link as well, if the domain authority was higher than the PR7 site.

What I'm saying is that it seems there more credibility to pages with higher domain authority than Page Rank.

From what I'm reading it seems like Page Rank is dead. or at least counts for only 5% of ranking algorithm today.

reference page site: http://www.seomoz.org/blog/what-is-googles-pagerank-good-for-whiteboard-friday

So what I'm getting at is DA>relevance when it comes to ranking.

Although I believe it's nice to have  High Domain Authority / Relevance sites, it's hard to get on them when you're back linking to niche keywords in a world of big brand competitors or your competition is just spam linking(which is a factor why they are number #1). They are  linked to over 2k domains while we are close to 600.

I figure the more domains you are linked to the better.

The better domain authority, (regardless of relevance or Page Rank) will boost your rank.

I believe that Domain authority plays a bigger part in ranking than relevance. However, I do believe that Relevance will play just as equal part as domain authority in future google updates.

I am also open to constructive criticism.