How to interpret the term "standard" which is found in FAA's Obstacle Departure Procedures (ODP) and Standard Instrument Departure Procedures (SID)?
The definition of standard minimum visibility for departure is found in CFR §91.175 Takeoff and landing under IFR.

f) Civil airport takeoff minimums. This paragraph applies to persons
operating an aircraft under part 121, 125, 129, or 135 of this
chapter.
(1) Unless otherwise authorized by the FAA, no pilot may takeoff from
a civil airport under IFR unless the weather conditions at time of
takeoff are at or above the weather minimums for IFR takeoff
prescribed for that airport under part 97 of this chapter.
(2) If takeoff weather minimums are not prescribed under part 97 of
this chapter for a particular airport, the following weather minimums
apply to takeoffs under IFR:
(i) For aircraft, other than helicopters, having two engines or less—1
statute mile visibility.
(ii) For aircraft having more than two engines— 1⁄2 statute mile
visibility.
(iii) For helicopters— 1⁄2 statute mile visibility.

The AIM 5−2−8 confirms your intuition about climb gradients.

b. What criteria is used to provide obstruction clearance during
departure?

Unless specified otherwise, required obstacle clearance for all departures, including diverse, is based on the pilot crossing the
departure end of the runway at least 35 feet above the departure end
of runway elevation, climbing to 400 feet above the departure end of
runway elevation before making the initial turn, and maintaining a
minimum climb gradient of 200 feet per nautical mile (FPNM), unless
required to level off by a crossing restriction, until the minimum IFR
altitude. A greater climb gradient may be specified in the DP to clear
obstacles or to achieve an ATC crossing restriction.


Note that visibility minimums do not apply to Part 91 aircraft. Also note that standard takeoff minimums do not have a minimum ceiling while published departure procedures almost always have a minimum ceiling or climb gradient—and often have both.