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The following paragraph is from a wired.com article:

When I visited this past fall, SpinLaunch employees were still unpacking from the move. As we walked among giant sheets of steel, Yaney explained how his launcher will work. A centrifuge large enough to contain a football field will whip a rocket around in circles for roughly an hour, its speed steadily ramping up to more than 5,000 mph. The vehicle and its payload—up to 200 pounds’ worth of satellite—will experience forces that, at their peak, will be ten thousand times stronger than gravity. Once it’s spinning at launch speed, the centrifuge will release the rocket and send it screaming into the stratosphere. At the threshold of the cosmos, it will fire its engine for a final nudge into orbit.

However, Wikipedia says:

The mean orbital velocity needed to maintain a stable low Earth orbit is about 7.8 km/s (28,000 km/h; 17,000 mph), but reduces with increased orbital altitude.

So, even ignoring the inevitable loss of speed as it travels from the launch site on the ground to the edge of space, and assuming it is still going at 5000 mph, it will surely take more than a 'nudge' to place it in even low earth orbit, because an increase of speed to 17,000 mph is an increase of 12,000 mph. This seems more than just a 'nudge'.

So my question is, what does 'a nudge' mean in the context of rocketry?

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"Nudge" is not being used here in the rocketry context, but rather in a marketing context.

There's no way to throw a rocket from ground level into orbit without making a significant burn once in space. Even if you could throw it fast enough to maintain something like 7700 m/s once outside the atmosphere, you'd still be in a closed orbit that would bring you back to ground level; you need to do a "circularization burn" in order to reach a stable orbit.

In the case of Spinlaunch, as you note, the lion's share of orbital speed still needs to be provided by a conventional rocket. The sole function of the word "nudge" here is to obscure this point to make SpinLaunch's proposal look more attractive.

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    $\begingroup$ "At the threshold of the cosmos"! $\endgroup$
    – Organic Marble
    Apr 26, 2021 at 16:17
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    $\begingroup$ @MatthewChristopherBartsh It gets more clicks/views, more such stories in the future, and it's quicker and easier than an objective dissection of the claims and explaining the technical issues to the readers. $\endgroup$
    – Christopher James Huff
    Apr 26, 2021 at 19:22
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    $\begingroup$ @MatthewChristopherBartsh many science journalists have absolutely no clue about the topic they are writing about and just regurgitate whatever they are being told. $\endgroup$
    – eps
    Apr 27, 2021 at 15:02
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    $\begingroup$ Daniel Oberhaus is a staff writer at WIRED, where he covers space exploration and the future of energy. He is the author of Extraterrestrial Languages (MIT Press, 2019) and was previously the news editor at Motherboard. They must fill their magazine up every month with new stuff. When not enough new stuff happens in a month... Anyway, at least they didn't accidentally invent a new particle! $\endgroup$
    – uhoh
    Apr 27, 2021 at 15:11
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    $\begingroup$ @eps I wonder why that is? Is it not a very sought-after job? $\endgroup$
    – Matthew Christopher Bartsh
    Apr 28, 2021 at 17:58

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