Hey Nadine,
From what I looks like, when the campaign is created, the StartDate is saved as Date time according to the activity log in adwords.
For example:
I created a campaign today (Jun 13th) using adwords API and I put Jun 14 for the start date of the campaign. My account's timezone is GMT+02:00 Jerusalem
In adwords activity log I see this line: Start time is Jun 14, 2016 12:00:00 AM IDT.
When I fetch this campaign from the campaign_performance report API, I get June 13 in the StartDate.
When I fetch this campaign from the CampaignService API, I get June 14.
Looks like the report is converting that datetime into another timezone and returning the converted date. maybe EST? Jun 14, 2016 12:00:00 AM IDT is Jun 13 5:00 PM EST
I hope this helps.
Idit