### Summary

I use the `ansible.builtin.copy` module. Unfortunately, there is a combination of parameters where check mode prints `ok`, although when running in run mode, there is an actual change.

Unexpected malfunction is reproducible tested on `2.9.27`, `2.12.6` and `2.13.0`

### Issue Type

Bug Report

### Component Name

copy

### Ansible Version

```console
$ ansible --version
ansible 2.9.27
  config file = None
  configured module search path = ['/home/phoffmann/.ansible/plugins/modules', '/usr/share/ansible/plugins/modules']
  ansible python module location = /home/phoffmann/.local/share/virtualenvs/ansible-2.9-TCZdLugh/lib/python3.8/site-packages/ansible
  executable location = /home/phoffmann/.local/share/virtualenvs/ansible-2.9-TCZdLugh/bin/ansible
  python version = 3.8.12 (default, Apr  8 2022, 11:41:59) [GCC 9.4.0]

$ ansible --version
ansible [core 2.12.6]
  config file = None
  configured module search path = ['/home/phoffmann/.ansible/plugins/modules', '/usr/share/ansible/plugins/modules']
  ansible python module location = /home/phoffmann/.pyenv/versions/3.8.12/lib/python3.8/site-packages/ansible
  ansible collection location = /home/phoffmann/.ansible/collections:/usr/share/ansible/collections
  executable location = /home/phoffmann/.pyenv/versions/3.8.12/bin/ansible
  python version = 3.8.12 (default, Apr  8 2022, 11:41:59) [GCC 9.4.0]
  jinja version = 3.1.2
  libyaml = True


$ ansible --version
ansible [core 2.13.0]
  config file = None
  configured module search path = ['/home/phoffmann/.ansible/plugins/modules', '/usr/share/ansible/plugins/modules']
  ansible python module location = /home/phoffmann/.local/share/virtualenvs/ansible-2.13-v8S06Uvz/lib/python3.8/site-packages/ansible
  ansible collection location = /home/phoffmann/.ansible/collections:/usr/share/ansible/collections
  executable location = /home/phoffmann/.local/share/virtualenvs/ansible-2.13-v8S06Uvz/bin/ansible
  python version = 3.8.12 (default, Apr  8 2022, 11:41:59) [GCC 9.4.0]
  jinja version = 3.1.2
  libyaml = True
```


### Configuration

```console
$ ansible-config dump --only-changed -t all
```


### OS / Environment

Ubuntu 20.04

### Steps to Reproduce

```yaml
---
- hosts: localhost
  tasks:
    - name: create file
      file:
        path: /tmp/ansible_foo
        state: touch
        owner: '{{ ansible_env.USER }}'
        group: '{{ ansible_env.USER }}'
        mode: 0600

    - name: Copy file with permissions
      copy:
        src: /tmp/ansible_foo
        dest: /tmp/ansible_foo2
        mode: 0644
        remote_src: yes

    - name: create file
      file:
        path: /tmp/ansible_foo2
        owner: '{{ ansible_env.USER }}'
        group: '{{ ansible_env.USER }}'
        mode: 0600
```


### Expected Results

I expect task `Copy file with permissions` to print `changed` 

Expected Result:
```
TASK [Copy file with permissions] ********************************************************************************************************************
--- before
+++ after
@@ -1,4 +1,4 @@
 {
-    "mode": "0600",
+    "mode": "0644",
     "path": "/tmp/ansible_foo2"
 }

```
It prints the expected `changed` as soon as `remote_src: yes` is removed.



### Actual Results

```console
TASK [Copy file with permissions] ********************************************************************************************************************
ok: [localhost]
```


### Code of Conduct

- [X] I agree to follow the Ansible Code of Conduct
mode:
    description:
    - The permissions of the destination file or directory.
    - For those used to C(/usr/bin/chmod) remember that modes are actually octal numbers.
      You must either add a leading zero so that Ansible's YAML parser knows it is an octal number
      (like C(0644) or C(01777)) or quote it (like C('644') or C('1777')) so Ansible receives a string
      and can do its own conversion from string into number. Giving Ansible a number without following
      one of these rules will end up with a decimal number which will have unexpected results.
    - As of Ansible 1.8, the mode may be specified as a symbolic mode (for example, C(u+rwx) or C(u=rw,g=r,o=r)).
    - As of Ansible 2.3, the mode may also be the special string C(preserve).
    - C(preserve) means that the file will be given the same permissions as the source file.
    - When doing a recursive copy, see also C(directory_mode).
    - If C(mode) is not specified and the destination file B(does not) exist, the default C(umask) on the system will be used
      when setting the mode for the newly created file.
    - If C(mode) is not specified and the destination file B(does) exist, the mode of the existing file will be used.
    - Specifying C(mode) is the best way to ensure files are created with the correct permissions.
      See CVE-2020-1736 for further details.
  directory_mode:
    description:
    - When doing a recursive copy set the mode for the directories.
    - If this is not set we will use the system defaults.
    - The mode is only set on directories which are newly created, and will not affect those that already existed.
    type: raw
    version_added: '1.5'
  remote_src:
    description:
    - Influence whether C(src) needs to be transferred or already is present remotely.
    - If C(false), it will search for C(src) on the controller node.
    - If C(true) it will search for C(src) on the managed (remote) node.
    - C(remote_src) supports recursive copying as of version 2.8.