Is it okay to use the cord from a Magsafe adapter with a different wattage? I have a 2011 Macbook Air.

The original Magsafe cord was getting flimsy, so I cut the old cord, got a spare one from eBay and soldered it to the power adapter.

However, I realised the wattage rating information built into the connector's chip is 60W in the new cord (which was probably obtained from a dead 60W adapter), as reported by "System information" and CocoaBattery, while my adapter's actual rating is 45W.

So far the Air seems happy with this new cord, but I wonder what risks this might incur, especially since 60W adapters are supposed to deliver 16.5v, instead of the 14.5v the Air needs.
If I'm reading your description correctly, then the replacement cord has a higher rating (60W) than the 45W that is drawn by the 2011 Macbook Air. This is fine because all it means is that the cord has a thicker gauge to handle a higher wattage (than what you are are actually using).

The replacement cord would not affect the voltage, so your adapter would still be delivering what it is specified for, namely the 14.5 V.