Why is the Market-Clearing Price also an Equilibrium Price?
(This is referring both to the initial question and to the comment. I would respond to each individually, but I don't think I have the reputation to address the comment directly.)

Paraphrasing a bit from your question, you're right- an equilibrium price is one where no firm has an incentive to deviate in their pricing or production decisions, and demand is perfectly satisfied (again, given the price). While the exact "mechanism" that would push price down might differ depending on what market type you're considering, the common feature will be that there is always a marginal incentive to deviate down to the lower price.

In terms of perfect competition- as you mentioned, it can be seen as a race for different firms to undercut their competitors, until they hit the no profit condition. That pushes the market clearing price down to marginal cost, which is the equilibrium price.

For monopolists, they're looking to maximize profit, not necessarily revenue. But, as every firm does, they choose a price p to satisfy:

$$
max\{q(p-c)\} = \pi
$$

They will choose a price higher than the one found under a perfectly competitive framework; hence why the market clearing price, and equilibrium price, is higher under monopoly than perfect competition.

But if the monopolist were to set a price different from the one that satisfies that maximization problem, they are pricing, as you described, too high. That tells them when to produce more units at a lower price; until their profit maximization price is hit. That becomes the clearing price, and the equilibrium price for a monopolist.

I'm sorry if that was too repetitive or elementary. You're question is an excellent one, and a somewhat subtle point has to be made. If it wasn't clear, or you'd like it re-framed, let me know!