What the Apple II did iirc depended on what version of the ROM. I believe the early versions booted into the monitor, at which point you had to type the magic commands. But they could be upgraded to a "II+" which autostarted the disk or dropped to AppleSoft. (And of course the easiest way to get the facts is to be wrong on the internet :)
Or the magic command to load a program from cassette tape, which was all they had. My later impression is the original Apple II monitor was a horrible UI. So probably most early Apple IIs were upgraded to the "II+" ROM.
Integer BASIC was part of the original ROM. If I had to make a guess, it defaulted to the monitor since you could buy machines in a 4 kB configuration. Even though Integer BASIC was in ROM, some memory would be used by the interpreter and some memory was used for video. That wouldn't leave much memory for software. (Heck, 4 kB wasn't sufficient to support high-resolution graphics.)
Loading Integer BASIC from cassette was a thing, but that was after the autostart ROM was introduced since AppleSoft and Integer BASIC were not compatible.
Our Apple II+ booted to Applesoft without a floppy disk. I think it automatically tried to boot from the floppy first. I remember that the oldest models booted to the monitor, but don't think I ever saw it myself.
Loading Integer BASIC from cassette was a thing, but that was after the autostart ROM was introduced since AppleSoft and Integer BASIC were not compatible.