A supplier is sending me very private information. For security, we are using thedecryptor.com website to encrypt the message. The website provides public and private keys.
This website doesn't share any information about how the encryption / decryption happens and should not be trusted for that reason alone.
We agreed that he will sends me a public key before payment, and a private key after payment.
I don't know what you are supposed to be doing with the public key. Yes, you can test if the given private key belongs to the same key pair - if you'd have known the algorithm, but nothing prevents the other party to use a different key pair altogether.
I would like to know if it is possible to change the content of the encrypted message while preserving the same public key (i.e, by changing the private key)
Normally you would expect that a change to a different private key would result in an error while decrypting, but that's not necessarily the case for every cryptosystem. As we don't know the algorithm anything can happen. I don't even see the possibility of sharing a public key on that website.
Normally you would make sure that the other party signs the message with a trusted public key, e.g. using sign-then-encrypt.