Are merit goods private? public goods? quasi public goods?

From what I understand, public goods are non-excludable and non-rivalrous, like national defense or street lighting. Quasi-public goods are those that are partly excludable or rivalrous depending on the situation, like roads that are non-rival until congestion sets in. Merit goods, on the other hand, are usually defined as private goods that create positive externalities, so they’re under-consumed in a free market, with common examples being education and healthcare.

Here’s where I get confused. Take education. It’s usually seen as a merit good because it’s excludable (if you don’t pay fees, you don’t get it) and rivalrous (limited class sizes). But in reality, adding one or two extra students to a class often doesn’t reduce the quality of education for everyone else, so is it really rivalrous? And if a government decides to make education free for everyone, does that suddenly make it non-excludable? Would that push it closer to being a quasi-public good instead of a private merit good?

Healthcare feels similar. It’s rivalrous since hospital beds and doctors are limited, but if a country provides free healthcare, doesn’t that make it non-excludable? So does that blur the line and make it a quasi-public good, even though we always learn to treat healthcare as a merit good?

So I guess my main question is: are merit goods always strictly private goods with positive externalities, or can they overlap with quasi-public goods depending on how they’re provided?