Although a SOT-23-6 MCU is the obvious approach, if you want to avoid the toolchain and coding work involved in using one, then the following timing diagram illustrates one approach:
pwr is a power-on flip-flop that is clocked by clk. It is reset by a power-on event.
long is a one-shot with a long time period that represents your long-...
I think the best way to modify your schematic to do what you want is to throw away most of the parts and use LTC2954 push button controller. It is designed exactly for one button control, does debouncing for you, supports adjustable power on and power down times and works up to 26V. Oh, and it does not assert output when you first apply power to it.