Bitcoin is backed by the cost of electricity (and equipment) required to compute its proof of work.
Not correct, though people say this very often.
"Backing" when talking about a currency has a very specific, important meaning. It means that some party has promised to redeem that currency for some other good of value. Backing requires a backer, in other words. It requires someone's promise to back Item A with Item B. This is how paper money can have real value, because if it's backed by a government or bank or corp that will give you gold/silver for it, then it has that same value by proxy. That's backing.
Bitcoin is not backed by anything, nor anyone. It is a resource, a commodity, a good in and of itself.
The cost of electricity/equipment does not back Bitcoin whatsoever, just as the cost of dredging deep-sea mud from the ocean floor doesn't back the price of this mud on the open market.
I agree mostly. Electricity price does however set a floor on the cost of production, while we are in the "block reward" phase at least. Bitcoin cost of production cannot be overlooked but is currently insignificant given the size of the monetary premium in the current BTC price, and will probably remain insignificant unless there is a massive crash.
Cost of production is the reason BTC did not go to zero after the last crash but stopped around ~US$2. Think of it like the backstop ... way, way, way down there and it is of course tied to difficulty and thus popularity of mining, i.e. network adoption.
It's very simple, if the market does not see any value in Bitcoin it will be worth zero. The cost of production only means the miners need to ask that price in order to recover their inputs. There's no "floor" that prevents people from offering less than is needed to cover their costs.
What it seems like is that some people are using the term "backed" to mean what gives bitcoin it's value. I don't think there is any underlying THING that gives it value. It's value is that it is extremely useful as money and it's proving to be a lot more useful than any past physical currency because of it's decentralized nature.