The formula for WAR is complicated - because there is a lot of normalization (to get numbers measured on different scales to add up) - but the fact that bWAR and fWAR are calculated differently should INCREASE your confidence.
The two sites differ in WAR, because they have a couple of fundamentally different assumptions about player performance. That is great - better to have a couple of perspectives on this question than a stat like fielding percentage which tells you literally nothing helpful. One of the major differences between the two was the notion of what "replacement level" is - but they have come to some sort of consensus on that. This is big, as it at least puts the two on the same scale.
But you look at position players - WAR is essentially identical between the two. The differences are in defensive measurement, and neither of those differences are that huge. Pitcher WAR differs, but it differs because the two sites have different ways of trying to separate a pitcher's contribution to run prevention. Fangraphs fundamentally gives pitchers less control over batted balls in play Baseball Reference does. This explains the huge gap in David Price's season for instance. The key thing is - I don't think that the question is settled (how separable is pitcher and fielding performance in terms of run prevention), and it is worthwhile to see how tweaking the assumptions changes the picture.
The two versions of WAR ask fans to basically engage the numbers and why they are the way they are - the fundamental assumptions about player value, and why good stuff happens. Remember, RBIs are also a made up stat - arbitrarily assigning credit for a run scoring.
You are right to a certain degree about offense vs defense. Chicks dig the longball - so people will always talk about homers and hitting and such - there is an inherent bias there. At the same time, defensive measurement IS much flimsier than offensive measurement. So yes, if I see two players with close WAR and one builds it with defensive value (see Victorino in 2013), it would move them down the rankings for me a little bit. But the defensive value is there - and it is nice that we can at least bound the contribution. When Bradley stunk at the plate, we DID hear about his defensive brilliance - because we knew that the equation could work if he hit enough. But he was so dreadful offensively, that no defense could overcome it. Outfield assists are a great extra parlor trick Bradley has - uncommonly good for a CF - but outfield assists are generally fairly random and a function of opportunity. And in a sense, your best bet to get outfield assists is to have an above average arm, but not a great one (because then nobody would run on you).