In asymmetric cryptography, it does matter which key we use to encrypt and which key we use to decrypt. From a high level, we want everybody to be able to encrypt so a public key is used for encryption. But we only want a certain party to decrypt (usually the party who created the key pair) so a private key is used for decryption. In other words, in the case of encryption/decryption, it's decryption that is the critical operation since it exposes what we're trying to hide - the plaintext.
Signing is the opposite. We want to restrict who can sign so a private key is used for signing. Remember, signing is like claiming responsibility or approval for a contract. So only the relevant party should be able to sign. Otherwise, we're allowing for forgery. Verifying the signature is not critical. In other words, everyone should be able to verify a signature, hence we use a public key for signature verification. In a public key system, the fact that a private key can only belong to one party gives us the extra benefit of non-repudiation. That is, no party who signed can deny that later, since they're the only party who possess the private key