I would agree with the assessment of a bad battery, especially if the battery is over or nearing five years of age. Just the average life span of a battery. As described, it is doing one of the things which can go bad with a battery, that being dropping voltage when under significant load, yet showing just fine when not. As stated, get it load tested to make sure.
As for it not working while attempting a jump start, realize most available jumper cables on the market are not of sufficient gauge to provide the needed amperage to start a car. Normally, these are considered "booster cables", which provide some power during starting, but their main goal is to actually recharge the battery for a short time until there is enough power in the battery to allow it to start the vehicle. Since your battery already looks good when not trying to start the car, it's not going to absorb much more power. When you turn the key to start the car, the high load causes the batter fault, and the booster cables are not enough to get the engine going.