I'm guessing this is not for pro gym ring exercises given that the rings have to extend on their straps 3m (9.8ft) from the attachment point - not much space for that under a porch roof. This also means that you won't be creating great dynamic load on the supports (no way to pull high-G swings on shorter straps, I don't think anyway) and so the load would pretty much be static and equal to your weight.
As far as 2x6 loading capacity, short spans like 6' aren't listed in the Uniform Building Code's span tables. The closest is 8'-6" and 6' would support more, so you're safe using 8'-6". Two 2x6es on 24" intervals would support 40PSF (pounds per square foot) live load (i.e. load that comes and goes such as people and furniture) plus 10 PSF of dead load (weight of the structural elements) but we won't count the dead load allowance here because the 2x6 is still a part of the porch roof support.
So the area between two 2x6, 6' long, on 2' intervals would have to be able to support 2x6x40=480 pounds of weight. Actually, slightly more since the formula is for 8'-6" spans but it just gives you a little bit of extra load for just in case. Since you are going to hang both rings on one joist, you'd have to delete the allowance by 2, so you get 240 pounds of load <strong>uniformly spread</strong> across the 6' span. The 2x6 will deflect 6'/360=0.2" when fully loaded (span/360 is the deflection norm) which is not too bad and you probably won't notice.
Since you're still interested in gym rings I'm guessing you weigh less than the max :) . The weight of you suspended from the rings in the center of the joist is not exactly a uniform load but the formula is, again, for 8'-6" span, so the extra strength that the shorter span affords would even that out. Besides, the straps would have to be spaced much further apart than the standard 50 centimeters (1.6ft) if you can't have the full 3m length of the straps. You'd have to work out proper spacing yourself but whatever it is, it'll help with making this load look more like a uniform one.
In short, it looks like you should be fine!
Good luck.