If it was a strict trade-off, I would agree that the answer was somewhere on the middle. But it isn't. Most of the things that give you precision will also give you accuracy. The few things that let you exchange one for the other look too much like cheating, and seem to always have much greater impacts on accuracy than on precision.
This is absolutely false - digital clocks are a common example. Precise to the second (or millisecond, or more!) but only as accurate as their setting.
Calculations also give precision without accuracy, like my package of tortilla chips - 13oz (368.5g) They took a measurement or specification precise to the ounce, multiplied by 28.3495, and got 368.5.
Accuracy and precision are independent variables, and the utility of either is usually limited to the order of magnitude of the other. With either "accurate to the second with a precision of milliseconds", or "accurate to the millisecond with precision to the second", you should only report a measurement to the second (or possibly a more precise measurement with an error range).
Do they have different meaning to you? They are the same thing unless there is some scientific meaning that makes them different of which I'm not aware. i.e. Theory in Science means a different thing than what it means to a layman person.
In common usage, these words mean the same thing, but scientists often give words specific meanings in order to make their research and ideas less ambiguous.
Definitions are even more important in non-scientific fields such as philosophy or math where meaningful reasoning of abstract structures and ideas would be next to impossible without giving them concrete definitions and stating your assumptions.
Remember that definitions are arbitrary, so to understand an author's argument or idea, you must seek out the author's definitions.
Hopefully that clarifies some of the discussion going on in the comments here.
[1] pre·ci·sion: noun the quality, condition, or fact of being exact and accurate.
[2]ac·cu·rate (of information, measurements, statistics, etc.) correct in all details; exact. "accurate information about the illness is essential" synonyms: correct, precise, exact, right, error-free, perfect; More
But it's inaccurate by many hours because you haven't set it yet.
If you're target shooting, and all of your shots go through the same hole: that's precision.
If that hole is several inches away from the bullseye, it's precise but not accurate.