First, DRX cycle is the number of radio frames in the paging cycle. Larger the value of this parameter, lesser the UE battery power consumption. Whereas smaller the value of DRX cycle, increase in the UE battery power consumption. Short DRX and long DRX in RRC_CONNECTED are they same, the difference is just the DRX Cycle length.

In LTE RRC IDLE, the UE needs to monitor PDCCH continuously (every 1ms) for P-RNTI to listen to paging message and this drains UE battery power heavily. Instead, if the UE is allowed to monitor P-RNTI in PDCCH, only at predetermined period say every 60ms or 100ms, it will reduce UE’s power consumption. The process of monitoring PDCCH discontinuously for P-RNTI to listen to paging message during RRC idle state is known as Discontinuous Reception (DRX) in RRC_Idle.

Extended DRX has been introduced earlier. but I am not sure when it was implemented or included in LTE 3GPP  standards. You can take a look at this.

http://patentscope.wipo.int/search/en/WO2013019287

A Machine Type Communication (MTC) device may establish a perpetual connected mode and an extended discontinuous reception (DRX) cycle with a base station. The perpetual connected mode may reduce the amount of signaling overhead required by the MTC device. The extended DRX cycle may enable the MTC device to reduce energy consumption while maintaining the perpetual connected mode. In addition, network overhead may be reduced by decreasing the frequency of scheduled measurements and/or transmissions performed by the MTC device.