I put my hands on almost all Brother push-button machines (if you see anything not mentioned below – I would appreciate if you let me know). If you would like to share any pictures I do not have (with the credits posted in my Encyclopedia, of course), let me know.
I am excited to start exploring soon push-button machines manufactured by other companies.
If you are new to machine knitting on flatbed machines, and you just got a machine, consider the beginner-friendly projects I recommended to all my students and customers. These projects can be done on any machine with any gauge.
ARTKNITTER

ArtKnitter DX-1000, standard, push-button
ArtKnitter DX-2000, standard, push-button
BROTHER
Brother, for sure, made the largest number of models of pushbutton knitting machines. The pushbutton machines are, in my opinion, miracles of intricate engineering. They offer robustness, longevity, and a lot of possibilities. Just combine your imagination, patience, and skill. Pusbutton machines are excellent for absolute beginners, those who transition from hand-knitting to machine knitting, designers, and simply those who like to tinker with needles and stitches but can’t hand-knit. Brother push-button machines are also a bit narrower than their mode advanced counterparts (like punchcard- and electronic machines), making them more compact and lighter. Additionally, their end-caps are not plastic but metal (except the KH710 pushbutton model), and as a result, they travel better if shipped.61
4-push button knitting machines

Brother KH511, standard
Brother KH521, standard
Brother KH531, standard
Brother Profile 551, standard
8-push button knitting machines

Brother KH110, fine
Brother KH111, fine
Brother KH120, fine
Brother KH552, standard
Brother KH560, standard
Brother KH561, standard
Brother KH580, standard
Brother KH571, standard
Brother KH581, standard
Brother Profile/KH583, standard
Brother Profile/ KH585, standard
Brother KH587, standard
Brother Profile/KH588, standard
Brother KH601, standard
Brother KH705, standard
Brother KH811, also with punchcard-patterning capabilities, standard
EMPISAL
All Empisal machines with pushbutton mechanisms are of standard gauge.
Empisal KH70
Empisal KH88
Empisal KH90, 8 pushbuttons
Empisal KH91, 8 pushbuttons
Empisal KH305
Empisal Vistamatic KH531, 4 pushbuttons
Empisal Compumatic KH600
Empisal KH680, 8 pushbuttons
Empisal KH6850L, 9 pushbuttons
Empisal KM 700
Empisal KH880, 8 pushbuttons
Genie The Matic Knitter KH710, 8 pushbuttons
JUKI

To my current knowledge, JUKI manufactured only standard-gauge knitting machines.
JUKI K-861, 8 pushbuttons
JUKI K-871, One Touch Auto S, 8 pushbuttons
JUKI K-881, One Touch Auto S, 8 pushbuttons
JUKI K-882, One Touch Auto S, 8 pushbuttons
JUKI K-883, One Touch Auto S, 8 pushbuttons
JUKI KE-1200, “Hi-memory”, 12-pushbutton/12-hole punchcard combo
JUKI KE-1500, 12-pushbutton/12-hole punchcard combo
MITSUBISHI
Mitsubishi Color 7 MH-700
SINGER/STUDIO/SILVER REED
As of today, I am not aware of any knitting machines made by Singer/Studio/Silver Reed with push-button capabilities. I’ve seen some machines on the Internet marked as Empisal but with “KH” labeling. They look different than pushbutton Brother machines so it is possible that those are indeed made by the father company Silver Reed.
TOYOTA
Toyota 858 (pushbuttons+punchcards), standard
OTHER
Speed Knitter SH-700, Super Speed Color A, standard