I was counting until I see the last opened document, which is also when the document is ready to be written to. So it's still 0.5 seconds, the same as for Apple Notes.
> until I see the last opened document, which is also when the document is ready to be written to
This is not true, many apps (or maybe it's an OS thing) retain an image that is not functional, you should count until you actually type any symbol and see it appear in the app. This also differs between switching to a recently opened app (hot) and an app that's been closed/unloaded from memory, both of which matter for instant note taking.
No, I'm not switching between apps. I'm closing it properly, lock the phone, unlock it and reopen the app. There is a ~0.2 seconds sync in the beginning, then the document appear and if I tap in the document then write a character, it appears in the document. About half a second before I can tap in the document and get a cursor after cold start.
If I wait ~3 seconds after typing I can see that the change has been persisted to the sync servers and it appears in my Obsidian desktop application.
The locking is not relevant, it doesn't change whether your app remains hot/in memory or offloaded (maybe after a while, but not in itself), neither is switching between apps or opening an app from your home screen, so not sure what you mean by closing "properly"
And for hot start you shouldn't even need to tap in the document if you edited it before, the keyboard will already be opened, so the measurement is "time start, tap the app icon, tap any key, see a character appear, time end". This is a common workflow for app switching and copy&pasting from/to note app/other apps, so here even half-second delays are noticeable, and where Apple Notes shines (and only very few apps match that, but not Obsidian)
Then the cold start is where using other apps forced note app to be unloaded (or after a phone restart or "force swiping up closing" the app (think it has the same effect), here the performance difference between Notes and Obsidian is even more noticeable
