Chainsaw bar oil is just your standard plain old straight-weight motor oil, but it has additives to it to make the oil "tacky" so it tends to stay in place better on the hot bar & chain while in operation - it resists being slung off. (If you took slo-mo video of a bar tip you'd be able to see the difference in the sling-off of bar oil vs motor oil.) If I didn't already have about a gallon and a half of bar oil left over from my small engine shop, I'd probably just go down to local dollar store and if they didn't have a quart of bar oil (they often do until they are sold out) I'd just get some straight weight oil (or better yet, hypoid gear oil) but Just have to be aware that bar, chain and sprocket would wear much more quickly and couldn't handle deep, long cuts like ripping a big log very well.
That said, I really never even considered there to be a "Best" bar and chain oil - around here, the best oil was the lowest priced oil.. so often it would depend on a shop's buying power and margin tolerance (Buy cases of oil by the truckload factory direct, and tolerance for thin margins, you could sell out that truckload in a few months - when dealers couldn't BUY bar oil for under $6/gallon, this one guy was getting truckloads at $3/gallon and then selling at $9/gallon, and made better margins than local dealers could trying to compete with him...) No one cared what brand it was, they just cared about the bottom line price.