Read the manual it's normal. The computer retards exhaust valve timing when first started for 30 seconds or less to quickly warm up the catalytic converter for emissions, which for that brief time makes the engine sound like crap. This is because of the long tube headers (none of your other cars have this) cool the exhaust. It's the compromise Mazda made to be able to run 13-1 compression with 87 octane and super high efficiency. The engine is also louder than those other engines when warm because of the direct injection, which runs up to 3000 psi vs 50 psi and MPFI on your other old school engines. This is the price you pay for diesel like efficiency on 87 octane fuel and still be able to meet emissions standards.
