Formally, your first program is invalid. Defining a variable in header file and then including this header file into multiple translation units will ultimately result in multiple definitions of the same entity with external linkage. This is a constraint violation in C.
6.9 External definitions
5 An external definition is an external declaration that is also a
definition of a function (other than an inline definition) or an
object. If an identifier declared with external linkage is used in an
expression (other than as part of the operand of a sizeof operator
whose result is an integer constant), somewhere in the entire program
there shall be exactly one external definition for the identifier;
otherwise, there shall be no more than one.
The definition of i
in your first example is a tentative definition (as it has been mentioned in the comments), but it turns into a regular full fledged external definition of i
at the end of each translation unit that includes the header file. So, the "tentativeness" of that definition does not change anything from the "whole program" point of view. It is not really germane to the matter at hand (aside for a little remark below).
What makes your first example to compile without error is a popular compiler extension, which is even mentioned as such in the language standard.
J.5 Common extensions
J.5.11 Multiple external definitions
1 There may be more than one
external definition for the identifier of an object, with or without
the explicit use of the keyword extern; if the definitions disagree,
or more than one is initialized, the behavior is undefined (6.9.2).
(It is quite possible that what originally led to that compiler extension in C is some implementational peculiarities of tentative definition support, but at abstract language level tentative definitions have nothing to do with this.)
Your second program is valid with regard to i
(BTW, implicit int
is no longer supported in C). I don't see how you could get any linker errors from it.