I should have said that you need to adjust this value to your needs.

>     \override InstrumentName.X-offset = -1

You can move the InstrumentName to the left (-) and to the right (+).
Instrument names are right aligned with the left staff edge by default.
You can change that using either X-offset, padding or extra-offset.

<<
  \new Staff \with { shortInstrumentName = "staff" } { a1 \break b }
  \new Lyrics \with { shortInstrumentName = "lyr" } \lyricmode { a1 b }
>>

Cheers,
Joram

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