The PM cannot ask for a short extension The legislation itself prevents it.
Under the terms of the bill, Mr Johnson must request an extension to Brexit negotiations to the end of January next year unless he can secure a deal or parliamentary approval for no-deal by 19 October
The commons library has more insights that may be of interest.
The Bill as presented to the Lords contains the following, as far as I know there were no relevant amendments.
The Prime Minister must seek to obtain from the European Council an
extension of the period under Article 50(3) of the Treaty on European Union
ending at 11.00pm on 31 October 2019 by sending to the President of the
European Council a letter in the form set out in the Schedule to this Act
requesting an extension of that period to 11.00pm on 31 January 2020
If the European Council decides to agree an extension of the period in Article
50(3) of the Treaty on European Union ending at 11.00 pm on 31 October 2019
to the period ending at 11.00pm on 31 January 2020, the Prime Minister must,
immediately after such a decision is made, notify the President of the European
Council that the United Kingdom agrees to the proposed extension.