Defensive Minded Coach
A defensive-minded coach will improve your defence ratings. Defensive-minded coaches are somewhat more effective than neutral coaches, who are in turn somewhat more effective than coaches who are fond of attacking, as discussed in the Hattrick 6.5 announcement.
As with any coach type, a team will typically prefer a defensive-minded coach either to help cover a team weakness or to enhance a team strength. Defence trainers or counter-attacking teams may prefer to enhance their defence ratings more, while more attack-minded teams may prefer to use a defensive-minded coach to cover for weaker defenders.
Advantage over Offensive Coach
Although the exact advantage of defensive-minded over attack-minded coaches is not stated in the rules or the HT 6.5 announcement, it is widely believed that the former are twice as effective. Any evidence for this claim would be appreciated.