I've been checking some online calculator (this one and this one) to determine the width of a PCB trace for a given current capacity, but I not sure if the value specified for current refers to direct or alternating current.
Do these calculators assume one type of electrical current or is the result applicable to both?
Both, AC and DC.
Your copper traces will fail by overheating, therefore it's the heating effect of the current that's important.
The calculator gives the RMS value of the current, which is the heating effect. For DC, this is just the same as the steady value. For AC, it's the square Root of the Mean of the Square of the current. A very 'pulsy' waveform, for instance the charging current from a bridge rectifier into the big electrolytic capacitors in a mains power supply, could have quite a high peak, but relatively low RMS.
The calculator must use the correct copper thickness. Note that at high frequency, the effective cross section of the trace reduces, due to the skin effect. Neither of those tools you've linked to appears to address this, so for high frequency (starting at several MHz for typical copper thicknesses) you would need to allow for this yourself. It's an unusual amateur design that uses high currents at high frequencies (ham radio transmitters perhaps), certainly forget about frequency for power supplies and audio.
