Anyone know why results in all the forms of Shogi on LG determine a single Shogi rating? I would have thought that standard Shogi and Tori Shogi, for example, require quite different experience bases to play well. And the smaller versions are largely tactical, whereas standard Shogi is substantially strategic as well as tactical.

I think it is, because the different Shogi games do not count as variations as for example 9x9 and 13x13 Go here at LG.
I agree with you that the various Shogi games require different skills. I am for example quite good at Minishogi, but I lack the strategic view and the opening knowledge at 9x9 Shogi.
I once asked Richard about this topic, but it seems to be difficult to implement.

I just checked my mail with Richard and it is the other way round. Go 9x9 and 13x13 are separate games and therefore have their own ratings, but Minishogi, Torishogi etc. are only variants and have no separate rating.

Go 9x9, 13x13, and 19x19 actually share ratings, even though they are shown as separate games on one's profile. (Look at my profile, for instance.)

Upon closer inspection, Hex and Reversi have separate ratings for the different board sizes. Go is just a strange exception in this regard.