The problem with that is that Hartley really doesn't have a system! Sure he plays exciting hockey, during a rebuild, but I somewhat question if it will ever be winning hockey. Most nights it seems like the average joe could do the same thing, everybody in and everybody back hard.
Hartley absolutely does have a system. Let me explain Sutter's system first, and then explain what Hartley does.
Sutter - Defencemen: Allowed to pass up along the boards, not up the ice. Seems safe, but resulted in other teams easily anticipating the pass and either cutting it off, or already closing the gap on the forward receiving the pass. It is a very 'safe' play, but it did result in the Flames getting hemmed in their zone too often.
Hartley - Defencemen: Defencemen were allowed to make a pass up the center ice, but with good judgement. Make a mistake, and they would hear about it. This made much more sense (not only because of the anticipation factor, but also the point below about cycling).
Sutter - Forwards: The system revolved around dump and chase hockey for the most part, with the goal in trying to establish a cycle. Sutter felt that the team could not score on the rush. While I won't argue that the Flames were a great scoring team off the rush, I would say that they were a worse cycling team. They were one of the smallest teams in the league, and one of the oldest. Not something you would consider beneficial to cycling. I also think that the runs the Flames went on were in part to opening up the game a bit and allowing the forwards to play more creatively. Sutter's system was a bit stifling, IMO, to creativity. The center (or one of the wingers) had to come back towards the blueline in case of a turnover as well. People regurgitated the "This team can't score off the rush" over and over again, but they weren't scoring off the cycle either.
Hartley - Forwards: Set plays are seem more important. His team still cycles (every NHL team has to cycle), but players are given the 'green light' to be creative and encouraged to score off the rush. The trade-off? If a player makes a mistake and causes a turnover, all the forwards HAVE to skate hard back to help the defence. That is why Hartley preaches fitness and runs tough camps. He demands that forwards skate hard to help put pressure on the opposing team's forwards with backchecking and defensive assignments. This is part of why Baertschi was benched the first time at least - so he could try and read how the other wingers were handling the defensive assignments under different scenarios. It wasn't because Baertschi didn't want to play defence (he was fairly committed IMO), he just missed assignments at a rate Hartley was not fond of.
Both coaches made a point to 'activate' the defencemen on offence, but was vastly superior under Hartley as the Defencemen there was used as a 4th forward trailing the play (usually), and since the Flames were not forced to generate offence off the cycle, this means they held possession longer, had their heads up to make plays, and found defencemen more often entering the zone to attack. You can see this clearly by how much better Bouwmeester seemed to be under Hartley than Sutter. I thought Jokinen had horrible IQ when I saw him consistently cross the blue-line, with a chance to cut to the net, but instead shooting it into the corner. When I saw Backlund start doing the same with an equal or even better chance at driving to the net, that is when I figured out that this was part of the system revolving around cycling.
These are just the 'basics' that I saw over the last few seasons. People who understand and pay attention to systems more than me can tell you much more about the nuances with each system (and even correct me if I was wrong about anything I wrote above).
Hartley does NOT utilize "River Hockey" as a system. It may look like that, but there are defensive assignments and expectations, as well as expectations and assignments on the offensive zone.