Let's have a look at lebensohl in the context of a weak NT. For what it's worth I put lebensohl (and 2NT in competition) amongst my 4 "absolute must have" conventions.
To start with what should we be doing when we have competition?
Firstly double: It seems to me that we need all the space we can get to rescue ourselves when we get doubled, and so I put "exit transfers" amongst my "5 most stupid conventions". Playing natural you have 6 calls available; XX, Pass, 2C, 2D, 2H and 2S. Playing exit transfers there is no useful meaning for 2S, so you have 5 calls available. You've cluttered your bandwidth with 17% noise. Keep playing them against me guys, but it's the ONLY convention I absolutely refuse to play with a PUP - I feel that strongly.
What about a suit bid? You can play double for take-out, but I find it's better to keep the pond-life honest, since they have this propensity to overcall on Queen-empty-5th and I need to teach them to stay out of my auctions, so I prefer my double to be penalties. 2-level bids need to be natural and to play. You can try 4-card suit bids here but it is a tad adventurous.
This leads us to what meaning can we ascribe to 2NT? Well it ain't natural, since over a double you'll want to play 1NT XX and your methods should permit this. Equally over a suit bid you'd have doubled with the natural 2NT, so there is no meaning for 2NT in a natural sense. This is the origin of lebensohl, and; like Topsy; it just grewed.
There are two species of hand which lebensohl can handle and I'll discuss them separately:
Firstly you might want to bid a suit - we can use 2NT to say "I want to compete in a suit, so just bid 3C and I'll pass or sign off. This allows you to bid 3-level suits naturally as a forcing bid.
[I'll digress here to look at rubinsohl, where 3-level bids are transfers. This is ok if you play strong NT because it keeps the strong NT hand hidden and by the nature of the auction the strong NT hand will usually be the stronger, but facing weak NT a natural forcing bid will be the stronger of the 2 hands and therefore you want the suit hand to be declarer, not the other way round.]
Secondly you might want to try for 3NT or 4 major. Here you need an agreement - FASS or FADS (Fast Arrival Shows Stopper or Fast Arrival Denies Stopper). This simply means that if you bid 2NT first in FADS you have the stopper but in FASS you don't. What you are doing is showing 4 hand types via a 2x2 matrix of bids. Lets look at the 2x2 matrix first:
first 2NT followed by cuing their suit or followed by 3NT
directly cuing their suit or bidding 3N directly
We can use the cue or delayed cue as Stayman; and we can use the direct bids as FASS or FADS. I prefer FASS myself as the auction 1N (2H) 3N sounds natural and I would be showing the stopper. With FADS, I'd deny it. You can choose but if someone says "Do you play leb?" then reply, "Yes, FADS or FASS?".
So here's the FASS table:
2N then cue: Stayman, no stopper
2N then 3N; Natural, no stopper
cue; Stayman and a stopper
3N; natural and a stopper
At this point you know whether 3N will be ok, and lacking a stopper you pull to your better minor to play. Equally if the cue is a natural minor then you bid 3H with both and can raise partner's 3S
The FADS table works like this:
2N then cue: Stayman, and a stopper
2N then 3N; Natural, and a stopper
cue; Stayman and no stopper
3N; natural and no stopper
Eclectic bidding theorists believe this is marginally superior, but the downside is when you forget, bid a natural 3N and the opponents cash 5 tricks in their own suit. I've lost more than I've gained by forgetting FADS which is why I play FASS.
Enough of the original lebensohl, there's another place where 2N in competition is very important.
Opponents open a weak 2 and partner makes a take-out double. You're going to assume he has the classic 4441 and about a 13 count, but often he won't have that and has stretched to find a bid, either on unsuitable shape or with significant extra strength. You can obviously bid sometimes at the 2-level and this is clearly a sign-off, but you may also need to bid at the 3-level and the problem once again is "Is it forcing?". We can use 2NT here too to solve the problem. 2NT says "I want to compete and play at the 3-level, so bid 3C and I'll sign off since, from your take-out double, you'll be suitable even for my 4 card suit." This releases a free bid at the 3-level as being limit values, and partner can move on if they have a bit extra for their original t/o. We still have the FADS or FASS structure in place too. Note that 2NT is not likely to have a natural meaning as with a double stop you'd be considering passing the t/o double.
It becomes evident that in competion 2NT almost never can have a natural meaning and most experts play 2NT as a puppet for 3C allowing them to play at the 3 level and consequently 3-level bids where 2NT is available are always constructive.
A simple rule is "lebensohl applies whenever the opponents have defined a suit of 5 cards or longer and we haven't yet got a fit". This works pretty well and I've seldom thought "I wish we didn't have this agreement" with my regular partners.
Well, there we go, another of my secrets revealed! Enjoy!