I understand that in the short run AS decreased (supply chains) and AD increased (stimulus, post-lockdown behaviour). But what's up with the lines in the long-run?
The image is hard to read because everything is crammed together and not properly labeled, but it seems to be perfectly fine description of an Covid19 lockdown effects using AS-AD. Graph shows the following 3 steps:
The graph shows that originally economy was at equilibrium given by intersection of the black short-run aggregate demand and black short-run aggregate supply and third blue line which represents long-run aggregate supply. Equilibrium was found at coordinates $(Y_0,P_0)$. The fact, that SR-AD, SR-AS and LR-AS intersect all at the same time indicates that this was original long-run equilibrium.
Next the graph shows that there was shift of the short-run aggregate supply to the left (red aggregate supply). This is perfectly fair description of what has happened, because lockdowns forced firms to restrict their production, and implement costly anti-covid measures, so that means at any given price they will be willing to supply less.
Next, the graph shows shift of aggregate demand to the right (red aggregate demand). This is reasonable at least for those countries that provided people with generous stimulus checks, since increase in income would shift aggregate demand to the right (although it might not be reasonable description for every single country).
We can see that the new equilibrium has coordinates $(Y_1,P_1)$. We can also see that this is short-run equilibrium, because long-run equilibrium should occur at some point where SR-AD and SR-AS intersect long-run AS.
The green AS, shows shift of AS to the right, again its reasonable that will happen in the future once the supply chains issues are fixed and lockdown restrictions lifted. It is reasonable to assume the pandemic would not destroy the long-run productive capacity of the economy. The green AD shows shift of AD to the left, I guess that is also reasonable once government will have to start withdrawing their stimulus spending.
Hence there is new long-run equilibrium, at a point where green SR-AS and SR-AD, intersect the long-run AS blue line. So the new equilibrium has coordinates $(Y_0,P_2)$.
Also is this even a valid way of showing AS & AD?
Yes, what other way of representing AS & AD did you ever seen? This is the most common representation of the AS & AD model that you will find in literally any good 101 macro textbook (e.g. see Mankiw Macroeconomics, Blanchard et Macroeconomics: European Perspectives, Krugman & Wells Macroeconomics).
I've never seen classical models like this.
The model is not classical model, it is Keynesian model, but Keynesian (respectively New Keynesian) and Neo-clasical synthesis (i.e. New Keynesian models with classical micro foundations) models dominated macro ever since publication of the General Theory by Keynes and you would find this AS-AD model in all major macro textbooks in last 70 years or so. If you google AS-AD and select show images then virtually all results will look like this.