He [Ed: Eve] then sends his half to Alice signed and establishes a connection. Now Alice is communicating with Eve and Eve is decrypting the message and sending to Bob.

Eve can only sign with her own private key. So your attack only works if Alice accepts the signature from Eve. That means that:

1. Eve's public verification key needs to be trusted and
2. that Alice doesn't check that that public key belongs to the intended party, i.e. Bob.

So no, this is not the same as raw Diffie-Hellman - unless the necessary verification steps are missing, of course.

He [Ed: Eve] then sends his half to Alice signed and establishes a connection. Now Alice is communicating with Eve and Eve is decrypting the message and sending to Bob.

Eve can only sign with her own private key. So your attack only works if Alice accepts the signature from Eve. That means that:

1. Eve's public verification key needs to be trusted and
2. that Alice doesn't check that that public key belongs to the intended party, i.e. Bob.

Some kind of PKI is usually involved, and PKIX is the most common PKI. But you can establish trust by e.g. sending the public key and then validating the fingerprint of it over the phone. I'm not sure if you can call that a PKI because of the simplicity of the protocol and the lack of other participants.

So no, this is not the same as raw Diffie-Hellman - unless the necessary verification steps are missing, of course.