How to interpret the term "standard" which is found in FAA's Obstacle Departure Procedures (ODP) and Standard Instrument Departure Procedures (SID)? This might be obvious, but I find it difficult to interpret correctly the term "standard", when it refers to the takeoff phase of the flight.

Take, for example, the takeoff minimums for the La Guardia airport, page L16. For Runway 13, it states 400' ceiling height and 2¼ statute (I think) miles visibility but it does not mention any climb gradient. Is it implied that it follows the "standard" climb gradient of 200 feet per nautical mile? Next, it states that alternatively one can takeoff with the "standard" visibility rules, but only with a climb gradient of at least 280 FPNM until it reaches to 500 feet altitude. Which exactly are these standard visibility rules? The only source I remember finding states a requirement for 1 statute mile visibility for 1-2 engines aircrafts and 1/2 statute mile visibility for aircrafts with more than two engines, but no requirement for ceiling height.

Another example can be found by looking at this SID. Its takeoff minimums description states that for runway 4 "Standard" takeoff minimums apply. In addition, for runway 13, it states "400-2 or Standard with minimum climb of 280' per NM to 500." Does this mean that for "400 feet ceiling height - 2 statute miles visibility" the "standard" 200' per NM climb gradient applies, while for "standard" visibility a 280' per NM climb gradient is required?

I would be really grateful if someone could help me clarify the above rules.
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.