LH moving into combat on 2nd or 3rd move

Yep Richard4th, that does appear to be the case so you are correct.
Page 9, second section from the top:-

SECOND OR SUBSEQUENT TACTICAL MOVES DURING THE SAME BOUND
Some elements or groups that have already moved this bound can make a second or subsequent
tactical move if there are enough PIPs and only if this does not start or go within 1 BW of enemy
(unless while moving along a road) and is entirely by:- (see also figure 5)
    (a) Light Horse or mounted infantry; and making a second or third move that is entirely in good going.
    (b) Psiloi making a second move either in their side’s first bound of the game, or if every element starts
         entirely in good going and ends at least partially in bad or rough going.
    (c) Troops moving along a road if making a second or subsequent move.

I can find no exceptions to this anywhere in the rules.

Note however that it is possible for an element to start the bound off-road,
make a tactical move onto a road, then make a subsequent move along that road.

DBA is a positional set of rules, where the location and orientation of our elements
tells us their current situation (in other words, elements have no memory of how
they got to where they are, just that they are there).

So providing the element starts the subsequent move on a road, they can use it.
Likewise, LH could start a bound in rough or bad going, make a tactical move into
good going, and since at the start of the subsequent move they are in good going,
they too can make a subsequent move.

Correct and the invader may choose any one of the four base edges as his own.

Ha!...here is another little trick for you Paul:-
    1) a LH element starts in a Threat Zone, pinning an enemy.
    2) it moves straight backwards out of that TZ using its entire move.
       (note: “using its entire move”, not its entire bound)
    3) since it's no longer in a TZ, it can now make two subsequent moves.
    4) and if it ends up on a road, it can make another subsequent road
        move to enter another Threat Zone to pin another enemy element.

...and all this in a single bound! (if it has the PIP’s of course)

Ave martin - as I understand that rule, the LH element may use some or all of its move distance.  The stipulation is that however far it moves that the move is restricted to being to a point directly to the elements original rear.  No turns or other deviation in direction are allowed.

Hopefully that is correct and clearer than mud...

Other players clearly roll much higher PIP scores than I do to make their LH dance like that!😉

Jim