Voyager EMH
Slipper Wearing Plucker of Pheasants
No.According to Law 12:
A direct free kick is awarded if a player commits any of the following offences against an opponent in a manner considered by the referee to be careless, reckless or using excessive force:
Therefore, by deduction, a push is not always a foul unless the referee determines it to be careless, excessive or reckless.
- charges
- jumps at
- kicks or attempts to kick
- pushes
- strikes or attempts to strike (including head-butt)
- tackles or challenges
- trips or attempts to trip
You have quoted from, "a direct free kick is awarded if"...
A deliberate push is a foul.
The referee may not always award a free kick for that foul.
If the referee's decision to award a free kick is the definition of whether a challenge is a foul, then the referee's decision can never be wrong.
In the recent incident involving Lloyd Kelly, he was pushed while heading the ball. This was reckless in my view, but was very clearly at least careless. The ref was wrong.