Assuming car battery is ok-ish. (Check this).
I see your button flashes amber.
This is from the manual doc. Two different procedures. The second is if the batt inside the key is low/dead.
(Forced engine starting method)
If the KEY warning light (red) illuminates,
or the push button start indicator light
(amber) flashes, this could indicate that the
engine may not start using the usual
starting method. Have your vehicle
inspected at an Authorized Mazda Dealer
as soon as possible. If this occurs, the
engine can be force-started. Press and hold
the push button start until the engine starts.
Other procedures necessary for starting the
engine such as having the key in the cabin,
and depressing the clutch pedal (manual
transaxle) or the brake pedal (automatic
transaxle) are required.
l When the engine is force-started, the KEY
warning light (red) remains illuminated and
the push button start indicator light
(amber) remains flashing.
Note:
When the shift lever is in the neutral (N)
position, the KEY indicator light (green)
and the push button start indicator light
(green) do not illuminate.
If the engine cannot be started due to a
dead key battery, the engine can be started
using the following procedure:
1. Depress the brake pedal (automatic
transaxle) or fully depress the clutch
pedal (manual transaxle).
2.Verify that the push button start
indicator light (green) flashes.
3. Touch the push button start using the
backside of the key (as shown) while
the push button start indicator light
(green) flashes.
4. After the push button start indicator
light (green) changes to full
illumination, press the push button start
with your finger.
- If there is a malfunction with the push
button start function, the push button start
indicator light (amber) flashes. In this case,
the engine may start, however, have the
vehicle checked at an Authorized Mazda
Dealer as soon as possible.
- If the push button start indicator light
(green) does not illuminate, perform the
operation from the beginning again. If it
does not illuminate, have the vehicle
checked at an Authorized Mazda Dealer.