Well, many of the bids after 1NT - (2H) aren't affected by Lebensohl. In particular, 2S, 3C and 3D are totally the same as if you weren't playing it - i.e. non-forcing at the 2-level and forcing at the 3-level.
Similarly, before Lebensohl was invented 3H would show 4 spades and 3NT would deny 4 spades, both with game-forcing values.
What Lebensohl did was give up the natural 2NT bid to allow the partnership 2 ways to get to the 3-level. So now you can show non-forcing hands in the minors and can distinguish between stoppers and no stoppers for 3H and 3NT. You also gain two ways to get to 3S and all the 4-level bids, but there are no standard meanings for those auctions, and partnerships have to agree on their meanings.
That's Lebensohl in a nutshell. Nothing to do with the double, and it's important to be able to separate the conventions you are playing (and, seemingly, teaching).
New topic - negative/takeout doubles
Back when Lebensohl was invented, pretty much everyone played double as penalty, so you could simply double the opposition with a hand like
xx
KQxx
xxxx
KTx
and all was good. The problem is people kept picking up hands like
KQxx
xx
xxxx
KTx
and you no longer have an easy bid when they interfere with 2H. Do you force to game? Lebensohl will help get you to the right strain and will avoid 3NT without a stopper, but this hand isn't really worth forcing to game. Do you bid 2S? Partner will expect five of them and you might wind up in a silly spot. Do you pass? You could easily lose out in the part-score battle.
Switching to takeout doubles on this auction solves the second hand but makes it harder to penalise them. This is a trade-off many partnerships are willing to make (all of mine included). The genuine penalty doubles don't come along that much, while having better ways to compete for the partscore is very valuable.
You don't entirely give up the chance to penalise them, since you can simply pass with the first one and hope partner holds a doubleton heart and makes a takeout double in balancing seat. If partner holds 3 of them, they are likely to find a better fit in one of the minors anyway, and then you can think about doubling them there.
Third topic - Lebensohl after (weak 2) - X - (P)
This is a separate situation to the one outlined above, and treating them as the same doesn't work all that well. But that discussion is for another time.
Other points
- Ron's last name is Andersen, not Anderson.
- Bidding has moved on enormously since Lebensohl was invented, and GIB reflects some of those advances. BBO and GIB don't lead the advances though.
- If you and your partner are forgetting this or any convention, you should probably think about not playing it.
- Lebensohl, in its various incarnations, is one of the better conventions. You gain a lot more than you give up, but you have to be prepared to put in the time and effort to learn it properly.
- There are better methods than Lebensohl, but mostly they give up the same thing - a natural 2NT - and gain the ability to better compete in partscore battles.