This varies from state to state and even within local boundaries. The following is from the MN Bicycle Handbook:

3. What is an electric-assisted bicycle and where can I use one?
Minnesota Statute 169.011, Subdivision 27:
“Electric-assisted bicycle” means a bicycle with two or three wheels that:
(1) has a saddle and fully operable pedals for human propulsion;
(2) meets the requirements:
(i) of federal motor vehicle safety standards for a motor-driven cycle in Code of
Federal Regulations, title 49, sections 571.1 et seq.; or
(ii) for bicycles under Code of Federal Regulations, title 16, part 1512, or successor
requirements; and
(3) has an electric motor that (i) has a power output of not more than 1,000 watts, (ii)
is incapable of propelling the vehicle at a speed of more than 20 miles per hour, (iii) is
incapable of further increasing the speed of the device when human power alone is used
to propel the vehicle at a speed of more than 20 miles per hour, and (iv) disengages or
ceases to function when the vehicle’s brakes are applied.
Minnesota Statute 169.011, Subdivision 47:
An electric-assisted bicycle with pedals and a motor that can’t go faster than 20 mph is
considered a bicycle under state law; all relevant bicycle laws apply.
Minnesota Statute 169.222, Subdivision 6(b)
The driver of an electric-assisted bicycle must be at least 15 years old.
Minnesota Statute 85.015, Subdivision 1(d) and Statute 85.018, Subdivision 2(d):
Generally electric-assisted bicycles may operate like bicycles do, unless otherwise posted.

In MN the high-powered go-fast ebikes are not considered bikes. I'm not sure how they are handled legally, but maybe MNDOT would have the answers. From what I have read an ebike that meets the MN standards can be used wherever a standard bike is allowed, unless a local rule prohibits them. I believe the "government standard" you refer to is probably a Federal standard, and I have heard that ebikes have been prohibited on some national park trails. But I have also read that they are allowed wherever a motor vehicle is allowed in some parks.

Either way, ebikes are growing, and there is always the "more power, more speed" crowd out there. That's why Dodge builds the Hellcat. Undoubtedly there will be some need for comprehensive rules and statutes. Fortunately I have not seen evidence of the abuse of ebikes in this area yet - I only encountered one high-power rig on the local trails in the last couple of years.