"to show somebody that you are interested in them and/or want to help them" That's a pretty vague definition, isn't it? By that, it could be argued that All of your examples are 'correct' uses of this expression. Reaching out to libraries on behalf of Google, for example, is an attempt to help make the information contained within more freely available. Even reaching out to an individual by cell phone is acceptable due to the preface of "If you want to follow up", which makes any act of doing so a 'sign of interest' and potentially a 'willingness to help out'. Unless, however, it's the "to show..." portion of your definition causing your Real hang-up. A Church reaching out to young people, using your example, is often a mere gesture; directed at no specific person or group. Compare this to the direct contact between individuals and/or organizations in your other examples and there's a clear difference. So if you ARE implying that "Reaching Out" ALWAYS has to be an indirect and empty gesture, I simply disagree.