This package P; our $a = 'something'; is equivalent to package P; $P::a = 'something';.

I don't believe this to be entirely correct since the second variant doesn't create a lexical.

the second variant doesn't create a lexical.

Neither does the first!

our only gives lexical visibility; not a true lexical variable.

Ie. With my, it creates an entirely new variable at each level of scope:

my $a = 123; 
{ 
    my $a; 
    print $a; ## produces "Use of uninitialized value $a in print ... 
+" because the above new $a is entirely new; thus uninitialised.
}

print $a;; ## prints: 123
[download]

However, our only gives lexically scoped access to a single variable:

our $a = 123; 
{ 
    our $a; ## produces ""our" variable $a redeclared at ..." referenc
+es the same variable as the first our $a
    print $a; ## prints 123 ## New scope, same variable, existing valu
+e.
} 
print $a;; ##  prints 123
[download]

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BrowserUk has already answered this question pretty clearly here and here. I just want to re-emphasize that all three statements you give as examples above do the same thing: they create (if the variable didn't already exist) and assign to a global variable $A in the P package (or namespace), with the our statement further granting lexical visibility.

The  $A = 'something'; statement is made illegal (due to all the foot injuries) by enabling strictures; see strict.