The IIe had a simple diagnostic routine, while later the //e with the 65C02 had a more complete diagnostic routines. But all three basically tested RAM, and the later tested some I/O. The II and II+ did not have this in ROM, though there was a diagnostic/repair floppy disk floating about somewhere.
The Atari XL Series also had a diagnostic test routines not found in the 400/800.
Machines that had programable character maps (Vic 20, C64, Atari), would also display graphics instead of text as their character map pointers which would say where the character maps would be, would be pointing to some random place in RAM instead. Later graphical machines like the Atari ST and Amiga would do like the Macintosh if they could not reach the point of generating a "Sad Mac" Code.As Elfen notes on a 6502 machine if there's a memory fault in the first 256-512 bytes of memory (the first page after the zero page is the stack) the machine will generally just wander off into the weeds and die. Usually the BASIC interpreter and low-level OS occupies a page or three after that that probably also isn't meaningfully tested (again, remember, all the memory test in these machines usually does is "count fingers and toes" to set top of RAM) so a problem there will result in a random crash or other symptom that isn't easily summed up in a screenshot. If there's one *common* thing about early 8-bit machines that indicates a fatal hardware issue it's a screen that looks like:
(Linked from Willegal's page about diagnosing and repairing Apple IIs) On old computers that don't require their video hardware to be initialized with software to display the contents of RAM if the CPU doesn't run (for whatever reason) you'll see a frozen page of random characters; when RAM chips are initially powered on their contents will usually be random, or close to it, and the garbage screen is the video generation hardware happily interpreting the contents. You'll see screens like this on broken/halted Apple I and IIs, Commodore PETs, TRS-80 Model I/IIIs, old CP/M machines, etc; on some of these machines if you look sharp you may actually see a brief flash of a garbage screen like this on a *working* machine, depending on how late in the ROM boot process it gets around to clearing the video memory. (TRS-80 Model I's with disk interfaces actually probe the disk controller and try to boot before they clear it so it'll be onscreen for a second or two; without disk it's cleared almost instantaneously.)
So far as it goes, under some circumstances broken Macintoshes will display garbage screens like this (IE, if there's an issue with the ROMs or something they won't even be able to generate a Sad Mac.); the difference is that Mac display systems don't have a "character mode", they're always displaying graphics, so a wedged/halted CPU will just result in a display of random dots instead of characters.