How to assign an interrupt to a button press with an ATtiny? (interrupt not fired with my code)
Look at the datasheet of the ATtiny45. In the section “Sleep Modes”,
there is a table that lists the wake-up sources available for each sleep
mode. For the mode “power-down”, INT0 is listed as a possible wake-up
source, but there is a small footnote:


  For INT0, only level interrupt.


This means that the CHANGE mode you are trying to use will not wake-up
the MCU out of power-down. You have to instead

attachInterrupt(0, wake, LOW);



  In my real code, I have 3 buttons, on pin #5, #6, #7, so eventually
  I'd like to have interrupts for all these three pins.


You may then have to look into the “pin-change” interrupt. This is
something different from the INT0 you are using here, and is not
generally supported by the Arduino core. You will have to either get
some pin-change-interrupt library, or work with no library using the
information from the datasheet (not that hard). Pin-change interrupt is
also a wake-up source for power-down.



Edit: Answering extra questions from comments.


  How did you know that "only level interrupt" means only "LOW"?


The Arduino documentation states that the mode
parameter of attachInterrupt() can be either LOW, CHANGE, RISING
or FALLING. A few boards also support HIGH, but the ATtinies are not
among them. A “level” can be either LOW or HIGH, whereas CHANGE,
RISING and FALLING are “edges”.

This is corroborated by the ATtiny45 datasheet: “The INT0
interrupts can be triggered by a falling or rising edge or a low level.”


  Is pin-change interrupt PCINT?


Yes.


  why do you think it isn't supported by the Arduino core?


The attachInterrupt() API only really makes sense for INT0, INT1,
etc... not for PCINT. You can also look at the source code.


  Isn't it standard to interrupt when a pin changes states?


If you wish, you can configure INT0 to interrupt on a rising edge, a
falling edge, or any logical change. But this uses an edge-detection
logic which relies on the main clock, and thus does not work in
power-down mode.

For more details on INT0 vs. PCINT, their differences, and how to use
them, please refer to the datasheet, specifically to the “External
Interrupts” section I linked to above.



Addendum: Extra question:


  why it doesn't work [with ISR(PCINT1_vect) {...}]?


Gcc tells me “warning: ‘PCINT1_vect’ appears to be a misspelled
signal handler”. And indeed, there is no such interrupt vector on the
ATtiny45: there is only PCINT0_vect, and this interrupt is shared by
all the pins that can generate a pin change interrupt. If you want to
know which pin triggered the interrupt, you will have to figure that out
within the interrupt handler itself.

This is the single point that makes the PCINTs trickier to work with
than INT0 and co.: each PCINT interrupt is shared by a group of pins. On
the ATtiny45, there is only one such group, and PCINT0_vect is the
only interrupt handler. Larger AVRs have more groups (3 on the
Uno/ATmega328P), and PCINT1_vect makes sense on them.

Also beware of the somewhat confusing naming scheme. Avr-libc defines
some macros PCINTn, where n is a number. These macros expand to bit
numbers, and are meant to be used in code like this:

PCMSK = _BV(PCINT0)   // sense changes in pin PCINT0 = PB0...
      | _BV(PCINT1)   // and PCINT1 = PB1...
      | _BV(PCINT2);  // and PCINT2 = PB2


In these macros, the number n identifies an individual pin. Avr-libc
also defines the macros PCINTm_vect for defining interrupt handlers.
In these macros, the number m identifies a pin group.