Man, this certainly got heated. A few observations:
It's showing the MS-DOS time/date prompt.
This is exactly the sort of burn-in you might get over the course of weeks/months if someone had a bad habit like, say, rebooting their computer when they were done for the day and left it on all night just sitting there at the boot prompts in a dark office. It's not going to happen if you leave the computer alone for a few minutes after flipping the power switch to go get a coffee or whatever once in a while. Ultimately I guess what you're asking for here is where on a continuum between "I left a prompt on the screen for a few minutes" and "I locked it in the closet with a C:> prompt on the screen and came back a year later" will *definitely* create an obvious and permanent mark. And the ugly truth is that nobody can give you an exact answer.
(I guess if you have a few copies of the monitor you're concerned about that you're willing to sacrifice to the cause you could probably come up with some experiments to determine that scientifically? Like, I dunno, dividing the screen into zones and writing software that cycles those zones at different frequencies... but even then the results are going to depend on variables like the exact brightness and contrast settings, etc.)
... Actually, I just looked at the picture of a 5151 on Wikipedia, and I don't think that's burn in. The green phosphor in the 5151 is so slow it'll hold an afterimage for a *long* time; I think it's very possible that program that was run to display that IBM logo was launched right before the picture was taken, and there wasn't enough time for the boot text under it to fade away after the screen was cleared.
I've seen
multiple sources stating monochrome CRTs are the most vulnerable to burn in. Due to a combination of the long-persistence phosphor and uniform intensity.
Yes, they're vulnerable to burn in, nobody disputes this, but ultimately it's not a phenomenon isolated to them; the same bad habits that'll burn a permanent stain into a monochrome monitor will also create "damage" on many other display technologies given enough time. (Because all human made artifacts wear out eventually.)
Also "modern" CRTs have an aluminum layer or something.
"Modern" in this context means post-1960 or so. Very old TV tubes were *very* fragile and it was quite normal for them to end up with a literal scorch mark in the middle of the screen.
Here's a patent from 1951 for an "Ion Trap", which was the standard solution for mitigating the damage in the 1950s, until someone came up with the brilliant idea of spraying a very thin layer of aluminum paint over the phosphor layer at the front of the tube to moderate the ion flux. Unless you have a very unusual monochrome monitor I'd assume this isn't going to be a problem for you.
About 1/4 of the monochrome CRTs I see on ebay have some burn in, if the seller bothers to plug it in. Obviously some of it depends on how it was used.
The first XT clone my family owned had a Princeton Graphics MAX-12 amber TTL monitor (a dual-mode monitor that supported both MDA and CGA in grayscale) connected to a CGA card. That monitor was in daily use for at least five years (many hours a day), and so far as I can recall it didn't suffer any significant burn-in for as long as we had it. But, again, a home environment isn't the same as an office or industrial setting.
Basically it boils down to this: if you spend eight hours a day for a few months or years doing nothing but programming in Qbasic then, yes, you'll eventually burn the menu bar into your monitor. If the monitor is doing *anything else* for reasonable periods then, no, probably not.