When you want a scene to look like someone filmed what is actually happening right now, not a staged scene. The Wikipedia entry on shaky cam says this technique gives a documentary feel, and gives as examples the handheld shooting of the first battle scene in Saving Private Ryan and parts 2 and 3 of the Bourne series directed by Paul Greengrass.
According to Wikipedia, cinematographer Janusz Kamiński shot the first battle in Saving Private Ryan handheld, showing the raw intensity of the Normandy landing from the landing craft to the beach, and shot most of the film handheld too.
To recreate it: Write the amount of shake too, like 'handheld, subtle natural shake', so the picture doesn't jolt excessively.
Wikipedia explains that parts 2 and 3 of the Bourne series, directed by Paul Greengrass, used a shaking camera together with fast editing, and that the average shot length is very short.
To recreate it: Cut clips where faces and hands easily distort, like crowd scenes, to 0.5 to 1 second to hide the weakness.
According to Wikipedia, The Bourne Ultimatum drew criticism for a shaky picture in dialogue scenes and for the listener's shoulder blocking the speaker. Film scholars David Bordwell and Kristin Thompson analyzed that Greengrass deliberately doesn't fit all of the action in the frame.
To recreate it: If you hang the shoulder or head of the person in front across one side of the frame in the start frame, it feels like secretly shot footage, and crowd faces show less.
In an American Cinematographer interview, Kamiński said he applied Technicolor ENR, a silver-retention process similar to bleach bypass, at about 70% to drain color and deepen the shadows.
To recreate it: Include only 'desaturated, high contrast' when generating, and apply the strong contrast in editing with the same value on all shots.
According to Wikipedia, Kamiński wanted the film to look like 'a major production shot by combat cameramen on 16mm', and he referred to WWII documentary films.
To recreate it: Include '16mm film grain, newsreel look', and if the texture differs from shot to shot, lay the same film grain over everything once more at the end of editing.
In the same interview, Kamiński said that much of the film was shot at a shutter angle of 45 or 90 degrees, and that 45 degrees was especially effective in the explosion scenes.
To recreate it: Include 'shot with a 45-degree shutter angle, crisp staccato motion, no motion blur', and splashing water and dirt don't blur but look choppy, bringing out the urgency of battle.
From 0:02 to 0:10 in the video, the Waterloo station scene catches the character from afar over the crowd, and blurry people passing in front block the frame and then pass.
To recreate it: Write 'long telephoto shot through a crowd, foreground heads out of focus' and you get the feel of surveillance footage watching from hiding.
Cinematic multi-shot sequence, 12 seconds total. Shot 1 (4s): handheld over-the-shoulder shot following a young female reporter with a press badge through a crowded rainy street protest, signs and umbrellas blocking the frame; Shot 2 (3s): handheld long telephoto shot from across the street compressing the dense crowd as she shouts into her phone beside a line of riot shields; Shot 3 (2s): handheld insert of trampled flyers and a broken umbrella on wet pavement as boots hurry past; Shot 4 (3s): handheld close-up of the reporter catching her breath, rain on her face, glancing off-screen. Consistent reporter and rainy street, flat overcast daylight, desaturated bleach bypass look, coarse 16mm grain, urgent raw mood.
A slightly shaking picture and half-hidden compositions look like footage that wasn't prepared in advance, which gives people the belief that it's a real event. Rough grain and drained color add the texture of recorded footage.
Handheld over-the-shoulder shot following a young female reporter with a press badge as she pushes through a crowded rainy street protest, signs and umbrellas blocking parts of the frame. Natural camera shake, overcast daylight, 16mm film grain, muted colors, urgent and raw.
Long telephoto shot from across the street, compressing the dense crowd as the reporter shouts into her phone beside a line of riot shields. Handheld camera with subtle shake, foreground heads passing out of focus, flat gray daylight, coarse 16mm grain, tense realism.
Insert shot of rain-soaked protest flyers and a broken umbrella trampled on the wet pavement as boots hurry past. Handheld camera with subtle natural shake, overcast light, desaturated tones, 16mm grain, gritty documentary detail.
Handheld close-up of the reporter catching her breath, rain on her face, glancing past the lens at something off-screen, framed slightly off-center. Subtle natural shake, flat gray daylight, desaturated bleach-bypass look, 16mm grain, exhausted resolve.
Join 1-to-2-second shots in fast cutting to create chaos, and use jump cuts that skip time within the same composition to give the feel of documentary footage that doesn't hide its editing. Slip in cutaways such as a flyer on the floor between shots to show details of the scene, which also covers sections where hands or faces got distorted.
Write the amount of shake, like "handheld, subtle natural shake", so the picture doesn't jolt excessively, and don't add any camera move other than the shake. Faces and hands in crowd scenes are easily distorted, so use them for only 0.5 to 1 second, or set up in the start frame a composition where the head of the person in front blocks the view. Put 16mm grain into the generation prompt, and unify the differing texture of each shot by laying the same film grain over everything once more at the end of editing.