There is a valid point in that this can be confusing.
Blender is not like an image editor (gimp/Photoshop etc) or many other programs where you work on one file and save it.
In one blender file you can create many new images for textures, masks, bakes and of course final renders, which need to be saved separately.
Internally blender will give new texture images a unique name/data block (untitled – untitled.001 etc) even if you do not save them.
If you add 3 image texture nodes and hit new in each one and do not give them names (leaving the default “untitled”) blender will add suffixes to keep the data blocks separate.
In the case of renders the name is “render result” and Blender always “overwrites” the “render result” data-block.
When using the image browser, with a particular image on screen, it would be more logical that when you hit save the name of that particular image’s data-block appears as the file name to save, instead of the name of the last image you saved.
Personally I always make a point of giving unique, valid, understandable names to all images, objects, collections etc But I can understand the frustration of the OP.
It is a bit like the Wait! You can't just press Render Animation thread, in this case Wait -you -cant-just-hit-save!