Lightning Experience: How to prevent the display of obsolete data?

Actually, this problem is due to the "aggressive" caching setting in LEX. Go to Setup Home > Security > Session Settings. Uncheck the option: Enable secure and persistent browser caching to improve performance. That should make it such that only a single page reload gives you new data/settings/etc.

To disable the setting mentioned in Greg's answer for scratch orgs, disable S1EncryptedStoragePref2 in your project scratch def JSON:

config/project-scratch-def.json

{
    "orgName": "My Scratch Org",
    "edition": "Developer",
    "orgPreferences": {
        "enabled": [
            "S1DesktopEnabled"
        ],
        "disabled": [
            "ChatterEnabled",
            "S1EncryptedStoragePref2"
        ]
    }
}