I picked the first option since it's closer to accurate, but the real answer is "neither."
First, I don't think "natural weapons" is a term in 5E, so you're kind of asking about the game-mechanical meaning of something with no game-mechanical meaning. That makes it harder to give a clear RAW answer. But, I think everyone here knows exactly what you mean, and as a concept I think "natural weapons" makes sense to most people. It's just that as a category, "natural weapons" covers two distinct things in 5E, not one thing:
1. Natural weapons that say they are unarmed strikes count as unarmed strikes, e.g. the aarakocra, lizardfolk, and tabaxi traits, and the effects of alter self. I mean, they use the words "unarmed strike" right there in the description. So they're usable with Martial Arts.
2. Monster actions that don't say they are unarmed strikes, aren't. So they don't work with Martial Arts. BUT... they're not weapons either, they're weapon attacks. Just like an unarmed strike is a weapon attack that doesn't use a weapon, a brown bear can use its Claws to make a weapon attack without actually using a weapon. So it doesn't work with Pact of the Blade either, nor can you target the bear's claws with magic weapon, etc. I can't find anything in the rules to suggest that the only two possible sources of weapon attacks are weapons and unarmed strikes, so it seems to me that "natural weapons that aren't unarmed strikes but not weapons either" should be a valid category of things that can make weapon attacks.
Now, if we want, we could get absurd and say that by this logic, a knight's Greatsword action isn't a weapon either, because it doesn't say it's a weapon. But since the word "greatsword" is literally a type of weapon, I think it's best to assume that the ability name conveys the additional information that, yes, this is a weapon, not just a weapon attack. So you could kill the knight, loot her greatsword, and attack with it. But if you kill a bear and loot his claws, you can't attack with them, except as an improvised weapon, because they're not actually a weapon, they're his fingernails. He can make a Claws weapon attack with them, but you can't.
OTOH, we could lean on "natural language" to rule that the bear is unarmed (we just established that he didn't have a weapon) and he's striking with his claws, therefore, by definition, he is making an unarmed strike. I think that's a valid interpretation, and it would allow the bear (or someone Wild Shaped into a bear) to use Martial Arts with the Claws attack. But I don't think that's the intention. Because by that same rationale you could claim that a monk's unarmed strikes were being used as weapons and are therefore usable with Pact of the Blade and other weapon-related spells, which can lead to some weird interactions, like applying alchemical silver to your fists. Which would I think would be a valid interpretation, and lots of fun for certain play-styles. But...
If you are looking for the best answer for "baseline, lowest-common-denominator D&D" then it seems to me that monster weapon attacks are neither unarmed strikes nor weapons unless they say so.
First, I don't think "natural weapons" is a term in 5E, so you're kind of asking about the game-mechanical meaning of something with no game-mechanical meaning. That makes it harder to give a clear RAW answer. But, I think everyone here knows exactly what you mean, and as a concept I think "natural weapons" makes sense to most people. It's just that as a category, "natural weapons" covers two distinct things in 5E, not one thing:
1. Natural weapons that say they are unarmed strikes count as unarmed strikes, e.g. the aarakocra, lizardfolk, and tabaxi traits, and the effects of alter self. I mean, they use the words "unarmed strike" right there in the description. So they're usable with Martial Arts.
2. Monster actions that don't say they are unarmed strikes, aren't. So they don't work with Martial Arts. BUT... they're not weapons either, they're weapon attacks. Just like an unarmed strike is a weapon attack that doesn't use a weapon, a brown bear can use its Claws to make a weapon attack without actually using a weapon. So it doesn't work with Pact of the Blade either, nor can you target the bear's claws with magic weapon, etc. I can't find anything in the rules to suggest that the only two possible sources of weapon attacks are weapons and unarmed strikes, so it seems to me that "natural weapons that aren't unarmed strikes but not weapons either" should be a valid category of things that can make weapon attacks.
Now, if we want, we could get absurd and say that by this logic, a knight's Greatsword action isn't a weapon either, because it doesn't say it's a weapon. But since the word "greatsword" is literally a type of weapon, I think it's best to assume that the ability name conveys the additional information that, yes, this is a weapon, not just a weapon attack. So you could kill the knight, loot her greatsword, and attack with it. But if you kill a bear and loot his claws, you can't attack with them, except as an improvised weapon, because they're not actually a weapon, they're his fingernails. He can make a Claws weapon attack with them, but you can't.
OTOH, we could lean on "natural language" to rule that the bear is unarmed (we just established that he didn't have a weapon) and he's striking with his claws, therefore, by definition, he is making an unarmed strike. I think that's a valid interpretation, and it would allow the bear (or someone Wild Shaped into a bear) to use Martial Arts with the Claws attack. But I don't think that's the intention. Because by that same rationale you could claim that a monk's unarmed strikes were being used as weapons and are therefore usable with Pact of the Blade and other weapon-related spells, which can lead to some weird interactions, like applying alchemical silver to your fists. Which would I think would be a valid interpretation, and lots of fun for certain play-styles. But...
If you are looking for the best answer for "baseline, lowest-common-denominator D&D" then it seems to me that monster weapon attacks are neither unarmed strikes nor weapons unless they say so.