Extending/Redesigning GPL code into LGPL lib: possible?

[zapro]

Some years ago I released a GPL'd application. Across the years, while
working on other projects, one of the files of this application outgrew
itself into a small yet independent library which I would like to
release with the LGPL license. Is this possible?

The new library can be considered an extension of that original file (I
wouldn't call that a library), in the sense that the main purpose
behind it is similar, but every aspect of the implementation is
different (to be precise, all function names / signatures are
different, and the implementations are also consistently different).

Going beyond my specific problem, I guess I am confused by the concept
of "different": to what extent two pieces of code can be considered
different in order to be published with at least "similar" licenses
(such as the GPL / LGPL), also provided the author is the same? (I
know, I know, the GPL and the LGPL are different...)