I'm going to try to write some sort of summary of this very complex topic that I've spent too much time on. Now we have information telling us how the different interactions work (mostly).
After all the comments in this thread, I've modified my views somewhat, and also Lantern probably has one of the most vague card texts in Dominion, so it's natural that we might read it in different ways.
I still think it would be most intuitive if Lantern and Way of the Chameleon worked the same.
We know that Lantern doesn't change (shapeshift) Border Guard, so that means it has to trigger when we play Border Guard, somehow modifying what we do. The only other cards that do this are Enchantress and all the Ways.
Enchantress and Ways trigger when you would resolve the play ability, and replace it. It's important to note that if one of them has replaced the play ability already, the other one fails. (This is what Ingix assumed early in the thread, but
Donald didn't confirm it until later.) In other words, you can't ever "override" Enchantress with a Way or vice versa, you just choose which to apply first, and the other does nothing.
If Lantern worked the same way, it would do essentially the same as Chameleon: Right before we would resolve the play ability, it's replaced with a version of itself. Then the question becomes,
(1) is it replaced with a modified version right now, or
(2) is it replaced with an identical version and the modification effects are set up to trigger on the fly? In the first case it would seem that Enchantress should fail. In the second case it's less clear, but it would seem Enchantress should succeed (in which case Chameleon would be an exception to the rulebook statement that Ways can cancel the Enchantment).
The first option is what I've been arguing for mostly, because then all the cards would have the exact same timing. But then I realized that the card text on Chameleon seemed to suggest the second option. The two options also matter for the -$1 token and -1 Card token, which trigger on when-would just like the modification effects would do in the second option.
But it turns out that Chameleon follows the first option: All the modifications are done before you start resolving, with the same timing as Enchantress. Enchantress fails because the ability is replaced, and this is the case even if the ability can't be modified by Chameleon (because of lack of +Cards/+$). And the tokens apply after Chameleon has done all the modifications.
When it comes to
Lantern, the ruling is that it works differently. It works as in the second option above, except it doesn't even do an initial replacement. When we play Border Guard and get to revealing cards, Lantern triggers and changes the number of cards, and when we get to discarding cards, Lanten triggers again, etc. This is what
Hhelibebcnofnena said in the beginning (except that you can't choose to apply Lantern's modifications before a Way).
There is the question of Chameleon with another Way, like Way of the Sheep, if you apply Sheep first. I would assume that Ways work like Enchantress: As long as the ability is replaced, Chameleon can't replace it.
There is also the question of Lantern with a Way like Way of the Mole. This is different, since Lantern doesn't replace the ability. It could be seen as, Lantern still triggers after you have applied Mole, because it's still a Border Guard, and it's telling you to discard cards. But based on
Donald's ruling, it seems that this is not the case: Even though we have a new discard effect, we're ignoring Border Guard's ability, so Lantern should never trigger.
Thirdly, there is the question of Lantern with Chameleon. Chameleon replaces the ability on Border Guard without making any modifications. This one is less clear, but I assume that Lantern is also like Enchantress in that it only works on the unreplaced ability of a card (in this case Border Guard). Lantern doesn't do anything with the whole ability (unlike Enchantress or Ways), but it does do something with specific effects in Border Guard's ability. This would mean that you can use Chameleon to play Border Guards "as-is" (to reveal less cards).
A way to view the card texts is then:
Chameleon:
"Instead of following this card's instructions: Follow this card's instructions with the exception that effects that give you +Cards this turn give you +$ instead, and vice versa."Lantern:
"When you would resolve Border Guard's effect of revealing 2 cards, instead reveal 3. When you would resolve Border Guard's effect of discarding 1 card, instead discard 2."