Checkbox conflict in my custom plugin admin page

you are registering two separate settings groups for setting fields & the problem might be here.

 function set_options() {
    register_setting( 'checkbox-1-settings', 'my_checkbox_1_name' );
    register_setting( 'checkbox-2-settings', 'my_checkbox_2_name' );
}

so use only one settings group for both fields to see if it works.

function set_options() {
    register_setting( 'checkbox-settings', 'my_checkbox_1_name' );
    register_setting( 'checkbox-settings', 'my_checkbox_2_name' );
}

Update

Get settings API and store in an array.

$options = (array) get_option( 'plugin_options' );

Now $options is an array containing all fields sub fields. and you can get the value of any key by just passing that key with this array.

for example to get a value of check_2

$check_two = $options['check_2'];