I'm drawing on variety of sources I've encountered working in and studying horticulture, and the answer is all the methods you've mentioned were used and varied situationally. Small areas might be maintained by simply trimming with hand shears or swops, or scythed with a specially chosen or very keen blade. Sheep were also an option for large less formal zones of grass as sheep graze in a nibbling fashion, cropping grass low to the ground producing a lawn like effect.

Once managing large areas of neat formal lawn became commonplace in the large landscaped properties of the UK, the Gang Mower was invented to make it easier. Think of a man-powered push along reel mower, but bulkier, and pulled along by a horse in concert with potentially two, three or maybe even five other mowers of the same type to cut a large swathe in one pass. Such mowers were the mainstay of large scale formal mowing until compact and reliable small combustion engines became available to produce equipment such as the Allen Scythe for meadow mowing, which would eventually evolve into what we would recognise today as a "modern" reel roller mower. Gang mowers were exported to the colonies to manage formal lawns there as well, and there were even examples designed to pulled by elephants for mowing lawns in such places where they were available as working animals.

I'm going to expand on u/Mrprocrastinaut's answer a little bit below, but he is essentially correct. Grass areas around 18th and 19th century estates would be maintained in a variety of ways. Very small areas might be hand-trimmed with shears, many areas would be scythed, and others would be grazed by animals. Different methods might also be employed at different parts of the same estate as well. One example (among many) would be Mount Vernon, George Washington's Virginia estate. Here he had a grassy "bowling green" directly adjacent to the house, which was highly visible, and which was maintained by scything and rolling. We know Washington was very particular about how this space was maintained, but it's difficult to say how it would measure up to the suburban lawns of the 21st century–How frequently was it cut? How short was it cut? How cleanly was it seeded? Washington purchased specific species of grass seed for both pastures and garden areas, but getting clean, consistent grass seed was challenging well into the twentieth century. 18th century gardeners fertilized grasses with things like ash and manure, but these were not as sophisticated as modern NPK fertilizers. So we know that the bowling green at Mount Vernon received a lot of attention from Washington's (free and enslaved) gardeners, but we shouldnt assume that it was maintained to the same specifications as landscapers today would use.

Adjacent to the bowling green was a much larger pasture, which was separated from it by a "ha-ha," basically a retaining wall with a ditch on the downhill side. This kept the sheep grazing in the pasture from getting onto the more intensively maintained bowling green, without creating any kind of separation, like a fence, that would be visible from the house. This is basically doing a little sleight-of-hand: the lack of visible distinction between the two spaces would lead observers in the house to infer, even subconsciously, that the much larger pasture in the distance was maintained in the same way as the bowling green. This gets to a larger point: there was not always a rigid distinction between lawns and pastures on large estates like Mount Vernon, and grass landscapes existed on a spectrum. There were rough pastures grazed by livestock without ever being seeded, cut, or irrigated. There were "improved" pastures, which had been seeded with specific pasture grasses, fertilized with ash or bone meal, and irrigated specially constructed ditches. These were still agricultural landscapes, but they might be placed in such a way to create a pleasing view from the house. There could be hay fields, which were allowed to grow long and then cut seasonally, leaving behind a uniform, lawn-like appearance that would sometimes be used for sports and games. Finally there could be lawns and bowling greens, maintained to a much higher standard and usually kept very close to the house. The techniques used to maintain these ornamental grass areas were variations of agricultural techniques–scything a lawn was basically just a variation on cutting hay. Animals were also used to maintain the grass in ornamental areas as well, and books on lawn-making from the early 20th century still mention this as a viable option for lawn care, even after the invention of rotary mowers.

EDIT:

I wanted to add on a response to u/rocketsocks comment above, where he states:

It's worth pointing out that because such landscapes required enormous amounts of labor to maintain they became symbols of high wealth and social status. As technological advances made them more affordable their status remained, so demand for them was quite high.

I want to push back on this answer a little bit–this is a common narrative and one that gets shared a lot, but I don't think it holds up really. Or at least, it doesn't get to the whole story. First of all, if you look at something like the gardens of Versailles, you see lots of labor-intensive elements that demonstrate the incredible wealth of the landowner. This include lawns, but also parterre flower beds, hornbeam hedges, labyrinths, topiaries, etc. Modern technology has made pretty much every one of these things more available to the average homeowner. So why is it that every house in suburbia has a lawn, but not a hedge maze?

I don't think there is one simple answer to "why do we have lawns," and there is a lot of history to unpack here. And certainly, when taste-makers like Andrew Jackson Downing were promoting lawns to Americans in the middle of the 19th century there is a class element to all of this. But I think using the master-narrative of "conspicuous consumption made widely available by technology" isn't sufficient. This narrative assumes that observers of a lawn in the 18th century would mentally process it as an absolute extravagance, something that required immense effort to maintain and served no purpose. The problem is, as I point out in my top level response to this question, grass landscapes existed on a spectrum from the purely agricultural to the purely ornamental, and ideas about pastures were mapped onto lawns and vice versa.

By the middle of the 19th century, for example, agricultural reformers in the United States had been promoting a version of English-style "convertible husbandry" for decades. This system kept much more land in grass pasture compared with the system in use in most of the United States. These reformers argued that grass-based agriculture was a more intelligent form of farming, and that keeping land in pasture was a sign of a sophisticated (and usually wealthy) farmer, contrasting him against the poor, somewhat desperate, frontier farmer who raises pigs and grows corn and tobacco. As a result, Americans in the 19th century already had a number of cultural associations surrounding grass pastures that they could map onto lawns. Quite tellingly, much of the early development of landscape gardening and landscape architecture in the United States had close connections to the sorts of agricultural institutions that had promoted convertible husbandry.

I think you also have to take into account the role that grass lawns played in the version of English-style "landscape gardening" that emerged in the 18th and 19th centuries. Compositionally, this style was all about figure-ground relationships, creating masses and voids to control views and sense of space. In this system, the "mass/figure" could be trees or shrubs of a variety of species, but the "void/ground" was pretty much always grass. The whole system relies on the existence of some kind of low, uniform, plant material; you can't do it without lawns, really.

There's so much more to say on this, but I just wanted to push back a little bit on the narrative you shared, because it is one that is repeated so often. It's not even totally wrong, it just feels like the kind of thing that a behavioral economist might come up with when trying to do history without ever really digging into historical sources to see what people in the past actually did or said.