The word online means controlled by or connected to another computer or to a network (Oxford).
The words query the oracle tells what you need and related to how your query is performed - online query and off-line query.
One queries the oracle then the oracle responds. Without a response, it is not an oracle. The response can be a bit or more, or even only return if there is an error ( wait x time, if not getting an error, you may assume it doesn't have an error).
The original padding oracle is executed online since the attacker doesn't have the encryption key and asks the oracle to check the padding is correct or not. If not correct, the oracle sends the padding incorrect error.
This attack must be online since the attacker sends ciphertexts to the test. The server ( oracle) has the key to respond to the status of padding after the decryption.
There can be a special case that creates the conflict; if an attacker has accessed your machine that has a program that responds to padding errors ( say the encryption and decryption can be executed in an HSM) then is this a local padding oracle attack or not? We can say still online since the attack created an online channel to execute the attack.
If you are just trying to program padding oracles, then you may write a program that only responds to the padding status, this is a local padding oracle that has no use. This will be used in online padding oracle attacks.
And, note that padding oracle is a decryption oracle that reveals the plaintext.
Partition oracle is also executed online, the attacker groups the keys produced by the KDF of passwords under a ciphertext and sends it to the server. The server responds tag as invalid or not.
Note that the partition oracle is a key revealing oracle since the attacker learns the correct password with less query than usual online password attack. Since the password generates the key, they reach the encryption key.
To better understand the oracle; consider a gnome sitting in a box with a book and a pencil possible with a coin, too. If an encryption oracle then a key is also included.
Now you query a ciphertext to the gnome and it decrypts and returns padding is incorrect or not. That is padding oracle.
Now you query plaintext then the gnome returns the ciphertext. This is an encryption oracle. If the scheme is probabilistic the coin is tossed for the IV then the oracle returns with the probabilistic encryption.
Now you query ciphertext then the gnome returns the plaintext. This is a decryption oracle.
Now, you query a ciphertext and the oracle returns the tag status. This is authentication tag oracle that partition oracle is used.
Now, you send a message, and the gnome checks the book for existence, if exist returns from the book, if not the gnome toss coins, write the query to the book, and returns the value. This is random oracle
Normally, the box is not on the attacker's access. If they can access inside then the key is revealed. So we assume that, while constructing the oracle, the attacker has no access to the box other than query.
So this is about how the query is executed. If the query is executed online we can call this online oracle, if not off-line oracle ( if this helps). In practice, online attacks ( remote) are more considered.
As a special case, the oracle can be a TPM chip on your PC. You are attacking this chip as research then this is offline. But an attacker can also attack this remotely, then this is online.