The Lord's 'Supper'

On Page 217 of Peterson he has a paragraph talking about the Lord's Supper that spurred me to think about the way in which we hold communion.

Would it be better for the church to actually eat a meal in keeping with the Lord's supper? This would encourage both a vertical and horizontal aspect of the Lord's Supper as Peterson encourages (p. 216), both an act of worship to God and an act of edification of the members of the body of Christ. All the different churches I have been in which I've had communion simply pass out the bread and the cup and then have a minute or two of solemness, reflection, prayer, thoughtful music, and then that's it. According to Peterson this misses half of the purpose of keeping the Lord's Supper.

Are our churches missing the edification of the body in the Lord's Supper. The whole body participates and identifies with Christ in this act. Why do we call it "communion"? Is it communion with God? Communion with each other? Both? And (this is the main question I am asking) should the church be having more of a meal together to celebrate the Lord's Supper instead of just a simple, short little time of remembrance?