The peak-load wiki page is pretty poor. Public transit may not be the best example, especially since at most times it's close to zero marginal cost but tends not to be free.
The bottom line up front difference is price discrimination is specifically intended to avoid charging an efficient price (in the basic microeconomic sense of market efficiency), whereas peak-load pricing is specifically intended to create an efficient price at all times.
Price discrimination explicitly segments customers by their personal demand curve, so that instead of offering the market price based upon aggregate supply and demand, thus admitting quite a bit of consumer surplus, a seller is able to offer exactly what each customer is individually willing to pay by charging a different price to each customer. Clearly, it's nearly impossible to do this in practice except by individual negotiation and secret prices, so you'll see segmentation into sub-markets rather than by individual customer, hence the airline example trying to use a fairly hacky means of capturing at least a little more producer surplus if not all of it. The point is to not charge a single market-clearing price, which inherently benefits customers who would be willing to pay more but don't have to. The producer itself is still willing to charge the lowest market-clearing price, but of course would prefer not to.
Peak-load pricing, on the other hand, has nothing to do with producers trying to optimize their own welfare surplus. A better example than public transit, sticking with public goods, is utilities pricing. A line can only sustain so much transmission bandwidth at a time and prices act as a rationing mechanism to induce usage to stay balanced over the course of an entire day so the line will always work. Charging more during the day is not intended to take advantage of customers who absolutely need the power at noon. It's intended to reduce peak usage so the line doesn't break.
Of course, it can work in the opposite way for goods that have short-term supply flexibility. Uber is the obvious private sector example. Peak-load pricing there is also not intended to gouge customers (in spite of the headlines). They do that so they can send bulk text messages to all their drivers who are not currently on the road telling them 'hey, sign on right now and you can earn ten times your normal fare.' It's intended to increase the system capacity to meet the level of current demand, while at the same time discouraging riders who are not willing to pay that much.
In both cases, the purpose of the price change is to always maintain a single price that actually clears the market. It's not that the producer is willing to accept a lower price but prefers not to in order to change the balance of producer's versus consumer's surplus. Instead, the price rises to prevent or remedy a supply shortage, and then drops again later to prevent or remedy a supply surplus.
