The pull-up, pull-down resistors forced ATX raspi to work with momentary button. Now MCU reads HIGH and works with momentary button as expected.
However, I noticed, that MCU probably gets some signals from the environment (or from the connection of power supply) and then "remembers" that HIGH/LOW state. For example, I took a new ATX Raspi fresh from the box. Once I connected it to the power, it blinks once and is ready to work with momentary button. Then I started to solder the connectors (power, USB etc). After each connector soldered I checked the status of LED by connecting power. Soon I noticed that it started to blink twice. Maybe this is because I connected power, disconnected it and then connected back right away. After my first attempt it blinked once, but after I connected power again it started to blink twice. From then on ATX Raspi stayed in the same "latching" state. So I soldered pull-up/pull-down resistors and ATX Raspi came back to momentary mode. Then I disconnected the resistors and it stayed in momentary mode. So this did some kind of reset to the MCU. Anyway, to avoid switching to latching mode accidentally, I think I will use ATX Raspi with pull-up/pull-down resistors from now on.