How is HEIGHT different from INHEIGHT?
(both don't exist and would be hard-fork changes)
INHEIGHT is the block the tx was included in (say #4). At that point in time HEIGHT and INHEIGHT both return 4.
100 blocks later HEIGHT will return 104 and INHEIGHT will return 4.
Introducing a new Proof of X system would require a hardfork to Bitcoin anyway - and is a
prohibited change. I'm more suggesting this for the theoretical implications.