You can shoot .223 in your WK 5.56. I have found that accuracy drops a bit in using .223 rounds. I used Winchester, Remington, S&B, Federal, PMC, etc.
I have found 62gr or 77gr 5.56 founds the best to use in my WK for some semblance of accuracy.
A gun chambered in .223 Wylde can shoot both.
For a pure .223 chambered gun, stick with just .223. Avoid shooting 5.56 in a .223 gun.
That being said, if you are unsure, check the barrel stamp where the caliber will always be stated and etched. Some imported guns that are sold here as .223 in ads are actually 5.56 (some export thing) - my Tavor X95 was like that - it was advertised as a .223 in the ad, but it is actually a 5.56 and it said that on the barrel and both North Sylva and IWI confirmed that.