do I need to give @EXPORT and @EXPORT_OK the name of the variable or the subroutine?

If I'm understanding you correctly, your question is whether you need to specify the same variable/subroutine in both the @EXPORT and the @EXPORT_OK array (?) If so, no, put it in either one of them.

@EXPORT_OK is for stuff to be exported upon request, i.e. when the user writes use Module qw(foo); while @EXPORT is for what should be exported by default (without explicit request), i.e. when the user simply says use Module;. If undesired, those default exports have to be disabled by writing use Module (); - note the (empty) list.

P.S.

require Exporter;

our @ISA    = qw(Exporter);
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can also be written as

use Exporter 'import';
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(see also Re: Advice on style)