Logic 101

If the fruit is not blue or the fruit is not red, do something.

Since the fruit can only be one colour, that logic will always return true.

>>If the fruit is not blue or the fruit is not red, do something.

The problem is when the fruit is blue, it still return true, but it has to return false

What is $name supposed to be? Your examples (!= or ==) are applying a completely different logic.

$name = "foo";

If $name is not "test1" => TRUE

This Condition breaks at this point. The second condition is never executed.

Consider the following Script:
-------------------------------------------
<?php

function first() {
  echo "first<br>\n";
  return true;
}

function second() {
  echo "second<br>\n";
  return true;
}

if(first() || second())
  echo "YES<br>\n";
else
  echo "NO<br>\n";

?>
-------------------------------------------

The output is NOT "first, second, YES" as you might think. It is "first, YES". 

The || operator checks whether either the left condition or the right condition returns true. If the left condition is true, the right condition is not anymore checked.

Daniel Lorch