More or less, but it's a little more complex if you want to flesh it out.

There are two major reactor designs in the USA (and most places), the PWR (Pressurized Water Reactor) and the BWR (Boiling Water Reactor). The PWR was developed originally for submarines, as part of the nuclear navy. The first PWR to ever "go live" for civilian use (at Shippingport, PA) was basically just a scaled up naval reactor.

The other kind of reactor, the BWR, is not a naval reactor. It was designed exclusively for civilian use. It was also developed by the government (Argonne National Labs).

In both cases, the US government invested in reactors both for civilian purposes as well as military (they are dual-use technologies — you can not only use them for naval propulsion, but you can also use them to generate plutonium). In the mid-1950s the idea was that the government would need to do a lot of the R&D for reactors both because industry wasn't sufficiently interested in it as a new form of power (indeed, it was a "boutique" power source, interesting but with difficult profit margins), but also because a lot of the field was classified (Eisenhower's policies declassified a lot of it by the late 1950s, and made it possible for industry to do work on classified topics, but there are a lot of regulatory considerations even beyond the secrecy issues).

So the PWR and Shippingport and other early reactors were basically meant to sell private industry on the idea of a reactor, and the fact that PWRs worked really well in naval contexts meant that you also had a lot of people who were trained in reactor operation coming out of the military (early reactor engineers in civilian plants were often ex-naval reactor operators). The push for industry involvement was explicitly an Eisenhower-era sort of approach — a free-market, Republican approach that wanted to divide the nuclear field into "peace" and "war" very neatly. Eisenhower's "Atoms for Peace" effort (started in 1953) was part of this, and also reflected Eisenhower's desire to make nuclear science something more than just a threatening weapon. As one can imagine drawing that peace/war line can get quite difficult with dual-use technologies.

There are millions of ways to make nuclear reactors — there are lots of engineering tradeoffs one can do depending on what characteristics you want them to have. They are immensely complex environments (as Perrow does a nice job of pointing out), with lots of mutually interacting feedback loops. So while there has been a lot of experimentation (Idaho National Lab, for example, basically is a reactor "test-bed" laboratory, and work has also been done at Argonne and Oak Ridge National Labs as well) with lots of different types, there is a lot of economic, organizational, and technological value in going with "tried and true." Similarly one can talk about why the US in particular committed to a uranium fuel cycle so early on (and not thorium) — it was what they were used to, and they already had a massive infrastructure in place for created low-enriched fuel (because of the weapons complex).

This is often known as "path dependency": you make one technological choice early on (for whatever reasons), and you end up getting "locked" into it down the line, because it's what your infrastructure (physical, financial, regulatory, and human) is set up to deal with. It makes it very hard to change paths later if you find that the conditions that led you to your early decisions no longer apply (ergo the thorium thing — it's non-trivial to start up a new fuel cycle from scratch, it is not as easy as just turning a switch). When I talk about this to my students I always have them try to imagine what it would mean if we wanted to ban all automobiles tomorrow in the USA, how difficult that would be, and why. It's not just that we use them constantly right now (and are in the habit of using them, etc.), it's that our cities are literally built around them, often to the exclusion of other means of transportation. (The degree of dependence varies — e.g. you can get away with not owning a car in NYC, but you really can't in most places.)

Which is why there are only a few reactor designs used in the world, and many of them go back to the specific military/industrial context of the 1950s. (The RMBK, one of the other major types, has its own interesting Soviet military/industrial context, having been built so that if desired it could be rendered into a plutonium-production reactor, and also having been chosen because its pieces could be manufactured by local factories as opposed to centralized facilities.)

On the push for civilian nuclear power and its awkward dual use aspects, see esp. Brian Balogh, Chain reaction: Expert debate and public participation in American commercial nuclear power, 1945-1975 (New York: Cambridge University Press, 1991). On the development of the PWR and its connections to Shippingport and beyond, see Richard G. Hewlett and Jack M. Holl, Atoms for peace and war, 1953-1961: Eisenhower and the Atomic Energy Commission (Berkeley: University of California Press, 1989), and Richard G. Hewlett and Francis Duncan, Nuclear Navy: 1946-1962 (Chicago: University of Chicago Press, 1974). Both of the Hewlett books can be found here.