This is a really broad question and the correct answer may actually depend on the regulations of a respective state for road maintenance. So I will try to answer this with references.
If you use the US Forest Service as an example, it has 5 road maintenance levels.
Road Maintenance Level 5
Assigned to roads that provide a high degree of user comfort and
convenience. These roads are normally double-lane, paved facilities.
Some may be aggregate surfaced and dust abated. The appropriate
traffic management strategy is “encourage.”
Road Maintenance Level 4
Assigned to roads that provide a moderate degree of user comfort and
convenience at moderate travel speeds. Most roads are double lane and
aggregate surfaced. However, some roads may be single lane. Some roads
may be paved and/or dust abated. The most appropriate traffic
management strategy is “encourage.” However, the “prohibit” strategy
may apply to specific classes of vehicles or users at certain times.
Road Maintenance Level 3
Assigned to roads open and maintained for travel by prudent drivers in
a standard passenger car. User comfort and convenience are low
priorities. Roads in this maintenance level are typically low speed,
single lane with turnouts, and spot surfacing. Some roads may be fully
surfaced with either native or processed material. Appropriate traffic
management strategies are either “encourage” or “accept.” “Discourage”
or “prohibit” strategies may be employed for certain classes of
vehicles or users.
Road Maintenance Level 2
Assigned to roads open for use by high-clearance vehicles. Passenger
car traffic is not a consideration. Traffic is normally minor, usually
consisting of one or a combination of administrative, permitted,
dispersed recreation, or other specialized uses. Log haul may occur at
this level. Appropriate traffic management strategies are either to
(1) discourage or prohibit passenger cars or (2) accept or discourage
Road Management Level 1
Assigned to intermittent service roads during the time they are closed
to vehicular traffic. The closure period must exceed 1 year. Basic
custodial maintenance is performed to keep damage to adjacent
resources to an acceptable level and to perpetuate the road to
facilitate future management activities. Emphasis is normally given to
maintaining drainage facilities and runoff patterns. Planned road
deterioration may occur at this level. Appropriate traffic management
strategies are “prohibit” and “eliminate.”
Roads receiving level 1 maintenance may be of any type, class or
construction standard, and may be managed at any other maintenance
level during the time they are open for traffic. However, while
being maintained at level 1, they are closed to vehicular traffic, but
may be open and suitable for nonmotorized uses
All of the attributes of the roads which fall under each level of maintenance and some example pictures can be found at Guideliness for Road Maintenance Levels
From this document, it does seem that even Roads which fall under Level 2 Maintenance are considered "maintained" BUT I would never rent a Toyota Camry and take it out on such roads; as it would only be logical to traverse such roads on a high-rise vehicle. There may even be road signs specifying what kind of traffic may proceed beyond a specific point (indicating the start of a Level 2 maint. road).
Another deduction may be that if you are on a level 1 maintenance road while it is closed you may be in breach of the TOS. (Although you might never be able to access it on car while it's closed).
I am not sure about private roads; whether states regulate the owners for maintenance or not, that is something specific to a locality and it would be best if you contact the local DMV for such questions. e.g. people have quarter/half mile long driveways in rural areas, what category would they fall in and what consequences would be for a car breaking down on them would be specific to the local law.
In general, this is just one aspect of the TOS the rental companies use to safeguard their litigation interests. Even in the statement that you quoted:
Generally, Hertz vehicles, including 4-wheel drives, are to be
operated only on regularly maintained roads. Off-road use is a
violation of the rental agreement.
covers two aspects of driving. One that the vehicle is to be operated on a maintained road. The definition of a maintained road in a certain area/jurisdiction will define where you can take the vehicle. Two Off-road is a violation. Now, in complete technical terms, off-road would mean where there is absolutely no road i.e. driving over sand dunes, through a creek etc. Are gravel/unpaved/dirt tracks etc considered off-road for the TOS? that's something you would need a lawyer for :-p
But all in all, such statements are a part of almost any TOS be it car rental, software, appliances, toys almost everything. The sole reason is to safeguard the company's interests in case of any legal action by a customer.
Furthermore, you would know that you are a safe driver and not have to worry about the TOS in such details. It's not like the rental company would actually track and record where you took the car to use the TOS against you e.g. you are returning the car and the person goes like just a minute sir you took the car to X area and breached the TOS you need to pay $$ amount or face legal action; that would be absurd. However, if you bring the bar back in a damaged condition etc then they would use each and every clause in the TOS and whatever else documents you signed to keep them on the safe side in legal aspects.