This is not completely true. This is, in general, something I've spent a good while looking in to...
In short (hah!), there are five ways a game can deal with saving data.
1) Don't save anything.
Save Lifespan: Until you turn the console off.
Used by: Most/all pre-NES cartridge games (I don't know of any console systems before the NES with saving... but I could be wrong, that isn't my strongest area),and many games on many platforms up until last gen. (Note that the DS is the first Nintendo handheld with saving in every game released for the system...)
2) Passwords.
Save Lifespan: Forever, as long as you wrote it down correctly.
Used by: Many games from the NES to GBA eras. Handhelds and cart systems see most of these games, like with no-save titles, but some CD system games are password-save only; some early PSX games, for instance (Warhawk, Pandemonium, etc).
3) Hard Drive/Floppy Disk.
Save Lifespan: Until the disk corrupts. Unlimited saves until then.
Used by: Famicom Disk System for the NES, 64DD for the N64, Xbox (hard drive), Xbox 360 (hard drive), Playstation 2 (hard drive), Playstation 3 (hard drive). PCs mostly use this system too, of course. Reliable... mostly.
4) Battery-backed memory (known as SRAM)
Save lifespan: Until the battery dies. Unlimited saves until then, but these batteries will probably only last 5-25 years and then will need to be replaced, not easy for most people. Replacing the battery (or having it die) wipes the cartridge of all saved data.
Used by:
-NES (all games with saving);
-SNES (all games with saving);
-Game Boy (all games with saving);
-Game Gear (all games with saving);
-Game Boy Color (most titles have batteries in them. All (I think) dual mode GB/GBC (black carts) have batteries. With clear GBC only carts, just look in them to tell.
)
-Sega Master System (all games with saving);
-N64 (all Controller Paks have batteries in them. About half of N64 games either don't save (some fighting games) or require a controller pak. Of the other half or so, most use EEPROM and a few use Flash RAM. About 12 US-released titles with on-cart saving, however, use SRAM battery save. These include F-Zero X, Ogre Battle 64 (though it also supports controller paks), Zelda OoT (but not MM, that's flash), The New Tetris, SSB, and a few others);
-Genesis/32X (most games with saving have batteries, but a few use EEPROMs - see below); -Sega CD (the internal backup ram and backup ram carts use rechargable batteries to keep their saves);
-Turbografx-16 (the two or three games with on-cart saving, as well as rechargable batteries that power the SRAM in the the save addon unit and the CD unit);
-3DO (system has internal battery, perhaps rechargable?);
-CD-i (again, internal battery in the system);
-Saturn (internal battery in the console, though it is user-changable and isn't rechargable);
-GBA (some titles use batteries for saving or for clocks. Most games have flash memory for saving, but not all.)
Note that all PCs, and consoles such as the Dreamcast, have batteries inside them, but these are used just for keeping the clock (and in PCs, for keeping the CMOS bootup information, or in Dreamcasts, for saving internet connection settings), not for saving data. Clocks can't run on flash memory, they usually need a battery (or internet connection, perhaps?). This is somewhat different, though.
5) Flash Memory (most common types: EEPROM, FRAM, and Flash RAM)
Save lifespan: Saves will stay on the cart for as long as the cart lasts (or at least a hundred years). The number of times the chip can be written to is limited, however, and varies depending on type from tens of thousands to a million or more. Once that point is reached you can't write to the chip anymore (but what is there will presumably stay).
Used by:
-DS (all games);
-GBA (GBA supports FRAM, EEPROM, SRAM (battery), Flash RAM, or no save. Without opening carts, it is hard to tell which games use which types; emulation sites list the types, but they can't tell FRAM (flash type) apart from SRAM, so the two are grouped together... making it impossible to know exactly which GBA games use batteries without opening all of the games in question and checking, I think.);
-GBC (specific titles have EEPROMS instead of SRAM. It's easy to tell which, because of the clear carts for GBC-only games. The only one I know for sure that uses EEPROM is Kirby Tilt 'n' Tumble.);
-N64 (Aside from the always battery-backed controller paks and the 12 (plus a few more Japan-only) SRAM titles mentioned in the above category, all other N64 games with on-cart saving have EEPROM or Flash RAM chips.);
-Jaguar (I think I heard that the carts are actually flash... not certain, though...);
-Neo-Geo Pocket Color (flash carts, all of them. Not all games have saving, though the ones that do are all through flash.)
Genesis/32X (all titles from Acclaim with saving use EEPROMs (NBA Jam, etc), mostly, but also a few others -- Sonic 3, Monster World III (Wonder Boy in Monster World), Monster World IV, and a few more have EEPROMs as well. The only companies to use EEPROMs on Genesis were Sega, Codemasters (Micro Machines series), Acclaim, and EA (for just one or two titles).);
-Saturn (plug-in memory carts are flash-based memory.);
-PSX, PS2, NGC, Xbox, Dreamcast, and X360 memory cards.
-Flash memory card formats such as SD, CF, Ministick, etc, as used by the Wii and PSP (and PS3).
-The Wii's internal flash memory.
-USB Flashdrives (if any consoles support them).
Unsure at the moment: Lynx (battery probably? not sure), Jaguar CD (I think I heard about this before, but forget at the moment what it uses), TG-16 memory cards (I would imagine that they use rechargeable batteries as does the internal save, but am not certain), all systems not released in the US, the Game.com, and possibly a few more.
For the GBA and N64 (among others), if you're wondering why so many systems are used, it's simple... size. Different types are different sizes and cost different amounts. For instance, on the N64 there were several options: no internal save (use no save or controller paks), 4Kb EEPROM (first in Mario 64), 16Kb EEPROM (first: Yoshi's Story), 32KB SRAM (Zelda OoT was one of the first), 128KB Flash RAM (Pokemon Stadium first). Most third parties saved money by just including controller pak saving; all games published by Nintendo have some form of save memory in them, as do some third-party games, but which type depended on how much space was needed. The GBA has different sizes available and adds one more type, FRAM, but the issues of space deciding which type to use remained.
I don't know the story why the Genesis had a small number of EEPROM titles and a lot of battery-backed ones, though... I wonder about it, but really have no idea. Listing of all EEPROM titles:
http://gxdev.files.wordpress.com/2007/12/gen_eeprom.pdf (
from this site )
Honestly, I think I prefer flash to battery-backed. Sure Flash RAM has a short lifespan, flash-wise (10,000?), but EEPROM and FRAM last longer, and any of them is likely to last longer than a battery, particularly because that battery continues to drain while it's sitting in the cart unused on the shelf, while a flash chip sitting in another cart won't be using up its writes as long as it's sitting there.