Particle Distribution
For the particle distribution, you can bake the Dynamic Paint (the wetmap to be precise) to an image sequence. The landscape object must be UV unwrapped for this to work.
Baking Step
create the UV map for the landscape object. It will take a minute or two because your mountain landscape has many vertices.
in the Dynamic Paint settings of the canvas (landscape object) choose Format: Image Sequence
and a higher Resolution, for example, 512
.
expand the Output section, enter a folder name for Cache Path. This is where the baked images will be saved. The press the Bake Image Sequence button.
Done this, you can deactivate the Dynamic Paint physics on the canvas and the brush with monitor and camera icons on the Dynamic Paint section in the Physics tab or alternatively, on the modifier itself.
Using the Baked Image Sequence
Use the baked image sequence for the shader to visualize it.
- choose Open on the Image Texture node and select all images with A. Make sure they are ordered by the file name to have them in the correct order.
- enable
[X]
Auto Refresh for the Image Texture node
- set the Number of Frames

Then create a Particle System for the landscape object.
- type is Hair, activate the
[X]
Advanced option
- use the cone (tree) object as rendered object (Render > Re[]nder As =
Object
and .. > Object > Instance Object = the cone). Increase the .. > Scale to a higher values like 0.2
.
- activate the
[X]
Rotation (advanced option), and set Orientation Axis to Object Y
to have the trees all vertically aligned.
- in the Textures section of the Particle System create a new texture in the first slot with the New button below.
Switch to the Textures tab.
- select the image sequence that you have created for the shader
- tick
[X]
Auto Refresh and check if the Number of Frames value is correct
- in the Influence section, tick [X] Density. And [X] Hair Length if you want to have the size affected.
- set Mapping to
UV
and select the correct UV map if you have more than one
(the particle system has a few old bugs. Sometimes it doesn't update and you have to do a change and revert the change immediately. Here for example are tiny trees... no idea why. Hair Length is not checked).
Geometry Nodes - Point Distribution
This is rather easy to do in Blender 2.93.5 because you can access the vertex color that is created by the Dynamic Paint physics directly in the geometry node:

The drawback here is that it is very slow when you replay the animation.
Related question
Is it possible to get a weight paint value for a vertex group inside of a material?