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I would just drop your former supervisor an informal email - he will know what to do.
And honestly, given that he knows you most likely very well the whole interview process might only be a formal procedure while the actual decision if it hires you again or not takes place in the moment he knows that you apply.
One other thought that you should keep in mind ...
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You are not required to stay at your university until you get a position in another PhD program. You can get a job instead. At the moment, they are plentiful.
Yes, you can ask for recommendations.
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This seems to be more of an "aspiration" rather than a rule. There is no enforcement mechanism possible, since it is the professors that submit the letters that control the process. It is probably also counterproductive in the sense that the "top researchers" in the field who would produce the best reference letters are also likely to be ...
1
Ask a couple of professors who best know your work and think highly of you. Tell them when you first ask that there will be several letters for different internships. If they know in advance then they can easily do it, even if each needs a bit of tailoring. Ask early but follow up a week or so before they are due.
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