"Way of eating" doesn't work for me. The other two are, unfortunately, too vague for me, but have become common usage.
To the point of usage, if no wood is involved it is not, and should not be allowed in any commercial venture to be called BBQ.
Jim - Another transplanted Texan
Former KCBS CBJ
Large and Medium Big Green Eggs , Black 18.5" WSM, Blue Weber Performer - Stainless, Green Weber OTG Kettle , Brinkmann SnP Pro, and a Stainless UDS. One retired Portable Kitchen grill.
Red Thermapen, Maverick ET-73, EdgePro Apex Sharpener.
Avatar is the original 1951 Weber Kettle