The starter is definitely suspect! After thoroughly checking the battery cable connections at both ends, I would use a voltmeter to see if there is voltage at the primary starter solenoid connection when turning the key to start. If the battery is properly charged and strong, and there is voltage present at the large cable to the starter solenoid, and present at the primary wire when turning the key to start, it has to be the starter.
One note here, I have seen more than one battery from AutoZone weak from the getgo. Sometimes they don't rotate their stock and you get one that's been sitting on their shelf for months and months. Those weak ones were common batteries too, not one with less sales demand like a Benz diesel battery. Just something to think about. Also I'm not throwing rocks at AutoZone, it's probably just our local AutoZone that is run that way.
'84 Euro 240D, 516K miles
'88 300E 5 Speed
Over 800,000 miles in