I'm going to give two responses to each, one being what I suspect might be Google's take on it, the other which is my take.
Which of these strategies, if any, cross the line from relationship building to link buying? Assume all links are do-follow.
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You're a local business. You give the local Boys & Girls club a few hundreds buck a year. In return, you get a very nice link on their Sponsorship page for 12 months. Google: paid; Me: paid
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You send a sample of your product to influential bloggers, for the purpose of a review and hopefully a link back to your website. Google: relationship; Me: relationship
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One of your clients is a college bar. You invite 50 college kids over for a slow evening and stuff them full of chicken wings. Then, you ask them to please review and link to the bar on their college wiki. Google: relationship; Me: relationship
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You give a client a free service, in exchange for that client linking to your business on its blog roll. Google: paid; Me: paid
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You take a blogger out to lunch, and pick up the tab. Later that day, the blogger writes up an amusing little story for the blog, and links back to your desired website. Google: relationship; Me: relationship
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In your email newsletter, you put out a request to your customer base, "Please link to my website, and I'll provide you a special 20% off coupon." Google: paid; Me: paid
Scary! Turns out I agree with Google on those... purely coincidence 
