Move to the `git_error` name in the internal API for error-related
functions.

Use the new object_type enumeration names within the codebase.

Next to including several files, our "common.h" header also declares
various macros which are then used throughout the project. As such, we
have to make sure to always include this file first in all
implementation files. Otherwise, we might encounter problems or even
silent behavioural differences due to macros or defines not being
defined as they should be. So in fact, our header and implementation
files should make sure to always include "common.h" first.

This commit does so by establishing a common include pattern. Header
files inside of "src" will now always include "common.h" as its first
other file, separated by a newline from all the other includes to make
it stand out as special. There are two cases for the implementation
files. If they do have a matching header file, they will always include
this one first, leading to "common.h" being transitively included as
first file. If they do not have a matching header file, they instead
include "common.h" as first file themselves.

This fixes the outlined problems and will become our standard practice
for header and source files inside of the "src/" from now on.

The `path_repository` variable is actually confusing to think
about, as it is not always clear what the repository actually is.
It may either be the path to the folder containing worktree and
.git directory, the path to .git itself, a worktree or something
entirely different. Actually, the intent of the variable is to
hold the path to the gitdir, which is either the .git directory
or the bare repository.

Rename the variable to `gitdir` to avoid confusion. While at it,
also rename `path_gitlink` to `gitlink` to improve consistency.

Error messages should be sentence fragments, and therefore:

1. Should not begin with a capital letter,
2. Should not conclude with punctuation, and
3. Should not end a sentence and begin a new one

Provide git_indexwriter_init_for_operation for the common locking
pattern in merge, rebase, revert and cherry-pick.