Strip the date out of your timestamp (I used strftime since I don't know what format your timestamp is) then add it to your by clause
source="source name" |search NameOfJob=$NameOfJob$ | spath "count.amountOfRecords" | search "count.amountOfRecords"=* | spath timestamp | search timestamp=*
| eval date=strftime(timestamp,"%Y-%m-%d")
| stats earliest(timestamp) as StartTime, latest(timestamp) as EndTime count by "count.amountOfRecords" NameOfJob date
| eval StartTime=substr(StartTime,1,25)
| eval EndTime=substr(EndTime,1,25)
| table date NameOfJob, StartTime, EndTime, count.amountOfRecords
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