When you save the image you see in the image editor, what you currently see in the image editor is what you get in the image you save (Or the top layer if you save as Multilayered EXR).
What the "Render Result" data block feeds from by default, is either the ViewLayer layer, or if you use compositing nodes: the Composite layer fed by the "Composite" node.

However, the ViewLayer layer is currently bugged, and saving to EXR defaults to the Combined pass no matter which pass you are currently viewing. It's already been reported in june 2024, and since then confirmed and added to the Pipeline I/O module's project board. Hopefully it will be fixed in the upcoming versions, but no plans yet.
In the meantime, you can use the Composite layer. Even if you don't do anything with the nodes, just having the default Render Layer node plugged into the Composite is enough.
To change what the Render Result sees, and more precisely to output the noisy image, you can plug the Render Layer node's Noisy Image pass to the Composite node, which will show that data in the render result, and let you save it manually:

An even faster way to quickly switch between different outputs and see the result in the image editor and save them, is to enable the Node Wrangler addon which allows you to ⇧ Shift
LMB any node to plug a viewer node to it (repeat to cycle thought the outputs), and open the "Viewer Result" in the image editor to see the Viewer node's results. This way you just have to point and click what you want instead of moving around a node and manually reconnecting everything all the time.
However, this is a manual method which makes no sense if you have multiple passes to save or, even worse, a sequence of images.
Hence, for saving passes (or even specific "steps" in your node tree), it is instead better to use either Multilayered EXR as output, which will save the final composite image plus all the Render Layer's passes into one file which you can retrieve in your editing software (example below in Affinity).

Or you can use a File Output node, and manually select different node outputs to turn into multiple files, even in different folders.
