As far as I can tell, the development team changed the way you access the start date early on Thursday. :) From the initial image, select the check box next to start date and you'll be able to enter a date.
With the integration of both start and due dates like this, you can access them both using the "d" quick key when you hover over a card.
The roll-out of the amalgamated buttons and the new pop-up window seems to have been staged across different regions. Yesterday, I had the two buttons on the card and two pop-ups; today I have the one button 'Due Date' and setting the Start Date is done on the resulting pop-up.
It's interesting that you can use the web interface set a date and time for when a card is due, but only a date for when it starts, but via the REST API you can set both a date and a time for the start.
Hmmm, this is the second question on the same day about the Due Date pop-up having different setting / appearances than expected. Refer to this question as well.
Your screen grab of the Change Due Date pop-up looks extensively different to mine:
The Date and Time fields are in different locations, have different names, the Save and Remove buttons have moved and changed size, the 'Prev' and 'Next' hyperlinks are '<' and '>' arrows, the table of days looks very different etc etc. That many differences can't be attributed to different page renderings by different web browsers.
For me, I can't set the Start Date via the Set Due Date pop-up. I have to push a separate button called Start Date to see a different pop-up that allows me to set that value:
It's possible the Trello team are experimenting with different code versions for the card buttons and the appearance in the pop-ups across different regions or Trello instances. Maybe try again next week and see what card buttons are visible and what is shown on the pop-ups.
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