(this answer is specific to the USA and FAA rules. Other countries may be the same, or similar, but I'm not familiar enough with them to guaruntee that this answer can apply outside of the USA)
To answer your question, we must first define what a "Standard Takeoff Minimum" is. Part 91.175 defines the standard wx conditions required for takeoff for commercial operators on an IFR flight plan:
§91.175 Takeoff and landing under IFR. (f) (2) (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.
Notice that there are no cieling requirements, only visibility. One- and two-engine airplanes must have 1 mile of visibility, and airplanes with 3 or more engines require 1/2 mile visibility.
Non-Standard - Some runways have obstacles in the distance or, for other reasons, require the weather to be higher than "standard" in order to use that runway during low visibility operations. These non-standard" minimums are published in the TPP. These minimums might be visibility-limiting only, or they might require both a minimum visibility as well as a minimum ceiling.
Ok, so that gets us to your question, in which you have provided specific examples.
Question: If take-off minimum depicts "Standard with minimum climb of 565' per NM to 2800, or 1000-3 with minimum climb of 370' per NM to 2800, or 1500-3 for climb in visual conditions."
- Standard with minimum climb of 565' per nm to 2800. This means that if your aircraft has the capability to climb at a rate of 565 feet per nm all the way to 2800 feet MSL, then you can use "standard takeoff minimums", which is 1 mile viz for 2 engines or less, or 1/2 mile for 3 engines or more, as long as you maintain that climb rate all the way to 2800'.
- or 1000-3 with minimum climb of 370' per NM to 2800. If your aircraft cannot maintain the climb performance listed in #1, but you CAN maintain a 370'/nm rate of climb, then you can takeoff when the weather conditions are 1000-3.
- or 1500-3 for climb in visual conditions. If you cannot maintain either of the two previous climb-out requirements, then you must takeoff in visual conditions and the weather observation must be a minimum of 1500-3
Question: If the airport has 1200-6, what take-off minimum can be used?
In this case, you have a visibility of 6 miles, which is higher than the minimum visibility requirements for all 3 items listed above. But you also have a ceiling of 1200 feet, which would prevent you from using Item #3 (due to the minimum ceiling requirement of 1500).
If your aircraft has the performance capability to meet either Item #1 or #2, then you have legally met the minimums and can depart.
The above information is only applicable to commercial flights operating under Parts 121, 125, 129, and 135 rules. "Takeoff Minimums" do not apply to flights operating under Part 91 rules. A part 91 flight can legally depart under 0-0 wx conditions. (although it is generally considered unsafe to do so, and several arguments have been made that it might be a violation of the rules to "operate safely"