Security of basic SHA256 MAC in an authenticate-then-encrypt scheme
Unfortunately your question doesn't fully specify the authenticated-encryption mode. In particular, AES is a block cipher, not a mode of operation, and only operates on plaintexts of some fixed length (128 bits for AES). So you need to fix a mode-of-operation for AES, such as CBC mode or CTR mode. (Never use ECB mode.) 

Let's assume you used CBC mode. Then the final Authenticate-then-Encrypt mode you outline is indeed more commonly referred to as MAC-then-Encrypt or MAC-Encode-Encrypt. The latter is a bit more accurate, as it captures the fact that often one needs to encode the concatenation of the message and message authentication tag as a plaintext string that is compatible with the encryption mode of operation. For example, CBC mode only accepts plaintexts that are a multiple of the underlying block cipher's block size (for AES this is 128 bits). 

In general MAC-Encode-Encrypt is never recommended for use in modern applications. Why? Because it's near impossible to implement correctly. One must first decrypt, then properly decode, and only then can check the MAC. The first two operations are done before checking authenticity and many clever attacks have shown how to confuse decryption implementations. See Lucky 13 for a recent example, and the included references to padding oracle attacks.