Is the y-axis always a spot price, an expected future price, or a contract price?
The y axis is not always any one of those. Typically the x-axis is a maturity date and the y-axis is the price of the futures that expire at that date, but some of these graphs use different axes values that make it confusing.
The basic definition of Backwardation is when the futures price is below the current spot price (technically the expected spot price at that time, but that's only different by the cost of carry - to make it simpler we're ignoring that). Visually a backwardated forward curve (price vs expiry) starts at the current spot price and is downward sloping.
What some of these graphs is trying to illustrate (poorly in my opinion) is the change over time of a backwardated futures price. If a futures price is below the current spot, the expected behavior is for the futures price to rise until it meets the spot price. I say expected in the sense that if the commodity future is significantly cheaper, then investors will "buy" the futures, bringing its price up. So the price of a backwardated futures is expected to rise over time, which is what the 2nd and 3rd charts are intending to show.
The last graph is essentially the same as the first, but they chose to have the futures cross instead of converge at the end.