You paste a screenshot into a note. It appears. Everything is fine.
Two years later you open that note and there is a grey box with a broken-image icon, or the picture is there but the database is 40 MB and you cannot work out why the app got slow. Both outcomes come from the same decision, made by the app on your behalf on the day you pasted, which almost no app tells you about.
The three things an app can do with a pasted image
Store it in the database. The bytes go into the app's own storage, alongside the note. The image is now part of your collection: it backs up when the collection backs up, exports when it exports, and works with the wifi off forever. The cost is size — your database grows by the size of every image, and screenshots are not small.
Store it as a file, referenced by path. The image goes into a folder and the note points at it. Efficient, transparent, and fragile in one specific way: move or rename the folder and every reference breaks. Some apps handle this well with a managed attachments directory. Some just write the path you dragged from, which means your notes now depend on a Downloads folder you clear monthly.
Upload it and store a URL. Common in anything cloud-backed. The picture is on somebody's CDN and your note holds a link. The note is tiny. The picture is not yours, it needs a connection, it is gone if the account lapses, and any export you take is an export of links.
The tell for the third is the export: export a note with an image and open the result somewhere else. If the picture is there, you have one of the first two. If you get a URL or a broken box, you have the third, and every screenshot you have ever pasted is a tenancy rather than a possession.

Why this matters more than it sounds
Images in notes are disproportionately the evidence. The error message. The receipt. The whiteboard at the end of the meeting. The before-and-after. The screenshot of the thing somebody said before they said they had not said it.
Text you can usually reconstruct. Evidence you cannot — that whiteboard is wiped, that error was transient, that receipt is in a bin in another country. Which means the images in your notes are, item for item, often the least replaceable content you have, and they are the part most likely to be stored as a link to somebody else's server.
What to do about screenshot bloat
The real cost of the good options is size, and it is manageable.
Screenshot the region, not the screen. ⇧⌘4 and drag beats ⇧⌘3 by a large margin in file size, and produces a more useful picture — a screenshot of a whole 5K display, viewed later in a note, is a picture in which nothing is legible anyway.
Crop before pasting, not after. Most apps store what you pasted regardless of how you have cropped it in the editor afterwards. The display shrinks; the bytes do not.
Do not paste the picture when you want the text. An enormous share of stored screenshots are screenshots of text — an error, a config, a message. macOS can select text directly out of an image in Preview and Quick Look, so lifting the text out and pasting that gives you something far smaller and, more importantly, something search can find. A screenshot of an error message is invisible to search. The error message as text is findable forever.
Check the database size occasionally. Knowing where your data lives is what makes this a two-second check rather than a mystery. If it has doubled in a month, it is images, and it is almost always five or six enormous ones rather than a hundred small ones.
Alt text, briefly
Write a line describing the picture, in the note, next to it.
Not primarily for accessibility, though that matters if you ever share the note. For yourself: an image is unsearchable, so the only way a screenshot turns up in a search two years from now is if there are words near it. "Screenshot of the deploy failing on the cert renewal, 14 March" costs eight seconds and is the difference between an archive and a shoebox.
It is the same discipline as writing the noun down in the first place. Nothing can find what you did not write.
The one to be careful about
Screenshots contain more than you meant to capture.
A screenshot of an error also contains the browser tabs, the notification that arrived, the name of the customer in the sidebar, the internal URL, and your own name in the corner. Pasted into a personal note on your own machine, that is fine. Pasted into a note that later gets shared, exported to a client, or synced into a workspace with other people in it, it is a small disclosure you never made deliberately.
The habit worth having: crop to the thing, before pasting. It solves the size problem and this one at once.
The honest version
If your work is genuinely visual — design references, a research corpus of images, screenshots by the thousand — a notes app is not your image library and you want something built for it. Managing tens of gigabytes of pictures inside a notes database is a bad plan regardless of how well the app handles it.
For the ordinary case of a few images a week as evidence and context, the only thing that really matters is which of the three storage choices your app made, and you can find out in two minutes by exporting one note.
Cyanote stores images in the same local SQLite database as everything else. They are in the backup because the backup is the database, they are in the JSON export because the export includes them, and they render with the wifi off because nothing about them was ever a URL. Notes hold images inline alongside headings, tables, callouts and code, and the clipboard history keeps copied images as well as text — so a screenshot you took an hour ago and lost is still recoverable from ⌥⇧V.