I don’t think this is great advice. College price is negotiable, but colleges like to feel like they are above the fray of commerce. As quoted in reply #3, some schools like Purdue flat out say they will not negotiate. So one practice is to negotiate without ever mentioning that word or otherwise making it plain.
When negotiating a FA offer for need, one can mention other schools’ more generous offer and ask them to “take another look to see if something was overlooked in their analysis”.
Let us know how it works out for you.