The "Light-water reactor" Wikipedia page states that "the uranium 238 atoms also contribute to the fission process by converting to plutonium 239". But from what I've read you need fast neutrons to breed plutonium 239 from uranium 238, not thermal neutrons. The point of using water is to moderate the neutrons so a LWH must be a thermal reactor by definition, right? So how can a thermal reactor, specifically a LWR, breed plutonium? Maybe it's actually some kind of hybrid thermal-fast reactor where some neutrons get moderated and some dont?
Heavy nuclides can undergo several nuclear reactions including neutron capture, fission, and scattering. The probability of these reactions are given by "cross sections" and are highly dependent on the neutron energy.
The fission cross sections for U235 (and Pu239) are much higher at thermal energies, so the goal of a light water reactor (LWR) is to slow the neutrons down to thermal energies to increase the probabilities of a fission reaction happening. For example, the fission cross section of U235 is about 500 times bigger at thermal energies than fast neutron energies.
The U238 fission cross section is a "threshold reaction", which means that it can only occur with high neutron energies. There is some U238 fission in a LWR, but it is not very much. The majority of U238 reactions in a LWR are neutron capture. During a neutron capture, the U238 becomes U239, then quickly decays into Np239, then quickly decays again into Pu239. Therefore, U238 does convert into Pu239 in a LWR. Pu239 is fissionable at low energies (like U235), so the Pu239 will eventually contribute to the total power. A rough rule-of-thumb is that at the end of an LWR cycle, 20% of the power comes from plutonium isotopes.
In a "fast reactor", the neutron energies stay in the high range and U238 fission can occur directly. However, there is still a large probability of U238 capture which eventually produces Pu239.
You need fast neutrons to fission U238.not to breed plutonium from it. The reason reactors don't use U238 as a fuel and fission it with fast neutrons is that fission of U238 would result in a runaway reaction which couldn't be controlled. To have a controlled nuclear reaction, you need delayed neutrons, which aren't present with fission of U238. As you say, fission of U235, the usual fuel of fission reactors, has to be done with slow neutrons, so the fission neutrons are slowed with a moderator. They can then be captured by U238 to transmute it into Pu239, as well as causing fission in the U235 content of the fuel rods. The daughter nuclides of U235 emit delayed neutrons at up to a second later, and this enables the reaction to be controlled.
Light water reactors are thermal reactors and can convert 238 U into 239 Pu. Technically they are "converters" not "breeders" because they cannot produce more Pu nuclei than they burn up fissile fuel nuclei. A fast reactor can be a "breeder"; that is, it can produce more Pu that it burns fissile fuel. Hence the interest in fast breeder reactors.
This behavior is because the average number of neutrons per fission reaction is greater in the fast reactor neutron energy range that in the thermal reactor neutron energy range, thereby allowing enough "extra" neutrons in the fast system to be available for breeding. Note that the fission cross section is greater in the thermal energy range than in the fast energy range, so thermal reactor fuel has less enrichment than fast reactor fuel; nonetheless there are more neutrons per fission in the fast than in the thermal fission process.
See textbooks on nuclear reactor theory by Lamarsh or by Glasstone and Sesonski.