Running yield and current-yield are the same calculation. The term shows up in gilt and European credit desks, and in property and loan markets where the point is the cash you collect per year rather than the total return.
Traders who fund positions care about running yield because it feeds directly into carry-fixed-income. If the bond runs at 5% and your repo funding costs 4.4%, you earn 0.6% a year for simply holding it.
Example: you buy 1,000,000 face of a 4.5% gilt at 92. You pay 920,000, collect 45,000 a year, so the running yield is 4.89%.
Related: current-yield, carry-fixed-income, coupon