Let me try to explain it.
0. Just set aside the distinction between real and nominal interest rates. My picture ignores that distinction, and so implicitly assumes expected inflation is constant. There's an interesting and important debate on whether central banks could escape a liquidity trap by increasing expected inflation. But that's not the topic of this post.
1. The economy is currently at the "we are here" point. The rate of interest (on safe short assets) is currently 0%. (I couldn't draw it exactly at 0% without messing up the picture, but ignore that). Real income is currently less than Y*, which you can think of as "full-employment income", "potential output", "NAIRU output", the "natural level of income", or "that level of income consistent with the economy being on its Long Run Aggregate Supply curve or Long Run Phillips Curve".
2. But the actual rate of interest (0%) is less than the "natural rate of interest" r*. That's what's different about my picture. In normal pictures of a liquidity trap, the IS curve is drawn sloping down, so the interest rate r* at which the IS curve cuts the "full-employment income" line is less than the actual rate if the economy is at less than full-employment income. The normal picture of a Zero Lower Bound liquidity trap has a negative natural rate. Mine has a positive natural rate.
3. Why does my IS curve slope up? It is not because I think that a higher interest rate would increase demand for newly-produced goods. Rather, it is because I think that a sustained increase in demand for goods would have such a strong self-reinforcing effect on desired investment and consumption that interest rates would need to rise to keep output demanded equal to output. The marginal propensity to invest plus the marginal propensity to consume exceeds one, in other words. Take a normal Old Keynesian IS curve, and assume mpc+mpi>1, and it will slope up. I explain this more in an earlier post.
4. I have drawn two LM curves. The horizontal LM curve is what is normally drawn in pictures of a liquidity trap. It is also the LM curve that is normally drawn by economists who think of monetary policy as setting a nominal interest rate. So I need to explain my vertical LM curve.
5. Take the Bank of Canada for example. In the very short run, you might describe the Bank of Canada's monetary policy as setting a nominal rate of interest. But that very short run lasts only about 6 weeks. What the Bank of Canada is really doing is targeting inflation. And a very important part of the Bank of Canada's monetary policy is its communications strategy. It tells people, at every opportunity it can get, that it is going to do whatever it takes to keep inflation coming back to its 2% target in the "medium term". It wants to focus people's expectations of monetary policy on that 2% inflation target, not on the overnight interbank lending rate over the next 6 weeks which, by itself, is nearly irrelevant to most people anyway. "In the longer run, the Bank of Canada sets the inflation rate at 2%" is a much better description of Canadian monetary policy than "for the next 6 weeks, the Bank of Canada will probably keep the interbank lending rate at 1%". And between that long run and very short run, what the Bank of Canada does is adjust the supply curve of reserves, and with it the overnight rate target, to try to keep income moving towards what the Bank thinks is "potential output", which is what I have called Y*.
The LM curve does not have a fixed slope. The slope of the LM curve is whatever the central bank wants to make it. The slope of the LM curve depends on how the central bank adjusts the supply of reserves in response to changes in interest rates, real income, inflation, and anything else the bank looks at when it decides what it needs to do to hit its target. And that depends on the time-frame. In normal times, I would say that the Canadian LM curve is vertical. It's vertical because the Bank of Canada makes it vertical. It's vertical because the Bank of Canada responds to IS shocks by doing whatever is needed to keep the demand for output equal to potential output, which it needs to do to prevent inflation moving away from target. A shift in the IS curve, with potential output unchanged, will have no effect on income. Simply because the Bank of Canada won't let it have any effect on income. That is equivalent to an LM curve that is vertical at Y*.
What about the Fed, right now? Well, we don't really know what the Fed is targeting. I'm not sure the Fed knows either, because different people at the Fed seem to have different opinions on what the Fed should target. Lacking any clearly-communicated strategic target, people can only focus on the Fed's short-run tactics. That's all there is. People have to try to infer the Fed's strategy from looking at its tactics. That's a problem, if the main way that monetary policy works is by influencing people's expectations of future demand, and prices, as well as future interest rates.
Here's my inference of the Fed's current strategy. It is unwilling or unable to communicate credibly any long-run target. If the recession looks like getting worse, it is willing to take what it considers to be extraordinary measures to stop it getting worse. But it is not willing to take those same extraordinary measures to make it better. I think the Fed currently has a sort of tipped-over L-shaped LM curve. The lower bit is near-vertical, and the upper bit is near horizontal. It wants higher than 0% interest rates, but it doesn't want the recession to deepen any further. So people expect the Fed won't let the recession get worse, but don't expect the Fed to make it better. Only if income increased back to full employment would the Fed's LM curve go vertical again.
But anyway. Look at the upward-sloping IS curve. Look at the positive natural rate of interest. Look at the 0% current rate of interest. Would you describe that as a "liquidity trap"?
My own answer is: well, you can call it whatever you like. But there is nothing there that says a return to full employment would require either we shift the IS curve (use fiscal policy) or else have negative interest rates. The Zero Lower Bound does not make it impossible to have an equilibrium at full-employment income.
And all that is leaving aside the question of whether we can escape the liquidity trap by reducing real interest rates by promising higher inflation.