How do I change the original (admin) user to standard user?

Login as the new admin account. Open System Preferences / Users & Groups. You might have to unlock this to make changes. Unlock the padlock in the bottom left. Then select the other admin account and the checkbox for Allow user to administer this computer should be ungreyed.

If you restart the computer, so all users get logged out, and then sign into the new admin account after a restart, can you change it?

Administrator: An administrator can add and manage other users, install apps, and change settings. The new user you create when you first set up your Mac is an administrator. Your Mac can have multiple administrators. You can create new ones, and convert standard users to administrators.

But you can't change the original administrator to standard .

Is the old Admin user still logged in? That might prevent disabling it.

FWIW, using a standard user account over an admin capable account only protects you from yourself. An admin capable account isn't root and doesn't have that power unless it elevates its privileges. The threat of a zero-day privilege escalation exploit is still a possibility, though very low.

See this article macOS Sierra: standard user

With the help of first original admin account , it is not be touched and its original name when the operating system was installed its name must not be changed , and with the help of it by clicking on plus sign in users and groups , you can either create a standard account or an admin account .

See this article how to convert standard user into admin account Set up users, guests, and groups on Mac - Apple Support

And if you have deleted the home folder of a user how to restore it back Restore a deleted user on Mac - Apple Support

And how to safely delete an admin account the last option is selected as home folder Delete a user or group on Mac - Apple Support

But in your older Mac I can not exactly presume why the admin account is greyed may its a refurbished machine and that account could be of previous owner .

Safer surfing (standard user) when doing "serious work."

Then virtually no benefit. It would only protect from a theoretical and exceedingly rare possibility of a privilege escalation exploit that can be triggered through the internet. Then, they have to have an exploit that can bypass System Integrity Protection (SIP) aka "rootless."