There are several different opportunities DAK offers to guests in regards to the safari. The following briefly describes them:
1. Sunrise Safari - Offered ONLY to concierge level guests at Disney's Animal Kingdom Lodge. Only offered on like two days of the week (I think Tuesday and Thursday but that may not be accurate). Offers a 45 minute to hour long safari aboard the standard truck before the park opens. Thus, the truck has the opportunity to stop at various points along the ride path. Will often see animals fairly active as they first shift out onto the safari. Does not offer backstage access as far as I know.
2. Wildlife Discovery Excursion - Offered to concierge level guests at any deluxe resort on Tuesday through Saturday in both the morning and afternoon. Will board a special smaller vehicle (with wheelchair access) accompanied by a safari guide and typically a member of Disney's Animal Programs. 45 minute to hour long journey into the Harambe Reserve with the vehicle often pulling off the standard ride path to allow the regular ride vehicles to pass. The vehicle typically will not stop unless it has reached a point that it can pull off the regular path (thus, it will typically not stop in the Ituri forest or Safi River portions of the safari).
The first two on the list are the ONLY offerings that allow for the vehicle to actually stop in the reserve so if you are wanting the best possible photo opportunities (personally, I just say live in the moment - but that is just me), these would be the programs for you.
3. Grand Gatherings (Wanyama) Safari - Allows for a slower journey (about 22-25 minute) through the reserve but does not guarantee stopping. 2 hour program that I believe includes dinner and a meet n greet with members of Animal Programs. Offered in the afternoon/evening - these are uaully the last trucks of the day that enter the reserve.
4. Backstage Safari - 3 hour program encompassing the backstage areas of Disney's Animal Kingdom and Animal Programs. Program includes an instructor from Disney's Education team who leads guests into various animal support facilities including the Animal Nutrition Center, Vet Hospital, Animal Programs Administration, and various Animal Night Barns. Typically also includes venturing back onstage to ride the Kilimanjaro Safari on a private truck to learn the secrets of the safari. However, truck remains on the ride path and follows normal show speeds (will only slow down for animals in the road as is usual). This program is probably the most varied of the four as it often changes due to the operational needs of the support facilities. The animal barns that are visited often change and there can be a good amount of variety. Also, there is no guarantee of seeing animals backstage because the vast majority of the animals are not held backstage during normal park hours.
I believe those are all the programs offered that include KSR. When booking them through Concierge or through the reservation center, be very clear as to which program you are looking to book because honestly, not all of the booking agents are entirely familiar with the differences.
Hope that answers some of the questions you may have.