I know that Natural Weapons don't all count as Unarmed Strikes, but I keep hearing online that Unarmed Strikes are a kind of Natural Weapon. Is this the case?
The damage from an unarmed strike is considered weapon damage for the purposes of effects that give you a bonus on weapon damage rolls.
An unarmed strike is always considered a light weapon.
Benefit: You are considered to be armed even when unarmed —that is, you do not provoke attacks or opportunity from armed opponents when you attack them while unarmed. However, you still get an attack of opportunity against any opponent who makes an unarmed attack on you.
A monk’s unarmed strike is treated both as a manufactured weapon and a natural weapon for the purpose of spells and effects that enhance or improve either manufactured weapons or natural weapons.
In regards to anyone other than a monk, an unarmed strike is a light weapon that is neither natural nor manufactured.
Exclusively for the monk (and others with monk-like abilities), an unarmed strike is a manufactured light weapon, and a natural light weapon.
What is a Manufactured Weapon?
Any weapon that is not intrinsic to the creature, such as a sword, a rock you pick up, a club you make.
What is a Natural Weapon?
Natural weapons are weapons that are physically a part of a creature, such as a bite, a claw, a tail slap.
Unless specifically mentioned, (such as the Monk, creature listing or racial description, etc.) no one has a natural weapon right away. For example, no human has a claw attack, a bite attack, etc. without feats, spells, or other abilities that gives that to them.
Also, an unarmed strike does not cause lethal damage, provokes an attack of opportunity, and is a very unappealing option without Improved Unarmed Strike, and/or some other way to make it better. Monk's specifically mention that their unarmed strikes count as both manufactured and natural or intents and purposes - because almost no one else's do.
If you take Skip William's advice to heart (which is not RAW) then you will find him stating, "For purposes of weapon enhancements, an unarmed strike is considered a natural weapon," and "Natural weaponry deals lethal damage without recourse to a class feature or feat, such as Improved Unarmed Strike."
While this is a common house-rule, it is not, in fact, the case, as you can see here, in wizards' introduction to the unarmed strike rules in 3.5.
The most relevant section is the following one:
Unarmed Attack/Unarmed Strike: These two terms are used interchangeably to describe an attack with an appendage that is not a natural weapon, such as a human's fist. An unarmed attack usually deals nonlethal damage and provokes an attack of opportunity from the creature being attacked.
It is a fairly useful house-rule, however, as many effects/upgrades/abilities that are otherwise limited to natural attacks already specify that they can be used with Unarmed Strikes as well and extending this to all such entities helps to make the Monk class, the only class likely to invest heavily in Unarmed Strikes, less underpowered.
It is important when making such a rule, however, to remember that the intent is to make unarmed strikes better, not worse, and thus to decide not to apply the restrictions, such as a lack of ability to make iterative attacks, inherent to Natural Weapons to Unarmed Strikes. Generally I have found the best such house-rule to be:
Unarmed Strikes count as Natural Weapons for the purpose of any effect, ability, or ability prerequisite that refers to Natural Weapons. They are still not actually Natural Weapons and thus are not subject to the rules for their use.