Voltage divider in combination with pull-down resistor
Let's redraw the schematic in a more common form:



simulate this circuit &ndash; Schematic created using CircuitLab
When the switch is closed, R3 and R2 are placed in parallel. Calculating the resulting total resistance gives us:
$$Rparallel =\frac{R2 * R3}{R2 + R3} = \frac{300k * 10k}{300k + 10k} \approx 9677 \Omega$$
The voltage divider output voltage then becomes:
$$Vout =\frac{Rparallel}{R1 + Rparallel} * Vin = \frac{9677}{100k + 9677} * 4V \approx 0,35V$$
Which is approximately the 0,4V you observed. So yes, your theory is correct.
However, according to the microchip website, the Attiny13 can work safely upto 5.5V. So you should be able to run it straight from the battery and it's inputs will always be able to safely accept voltages that lie within the supply voltage range.
In case you still want a voltage divider on the input, you do not need a separate pull down resistor. The bottom resistor of the divider will act as a pulldown when the button is not pressed if you feed the entire voltage divider trough the pushbutton:



simulate this circuit