By default, map layers use metadata from the data source metadata. A map layer does not have its own metadata. The layer can only be set to have its own metadata via the Properties dialog box. The drop-down list in the Layer Properties dialog box allows for this option, as described in the 'Edit content on the Properties dialog box' section in the help topic: https://pro.arcgis.com/en/pro-app/latest/help/metadata/view-and-edit-metadata.htm#ESRI_SECTION1_758736AA33AE4F2EA800839DF3C091E7.
Since the layer is not set to have its own metadata, the changes made to the layer's metadata in Step 3 are not displayed in Step 4. The layer is set to display the metadata from the data source. Changing the properties to show the layer's metadata instead of the data source metadata displays the metadata typed in Step 3.