All super funds – including SMSFs - listed on Super Fund Lookup have the ATO naming prefix “The Trustee for”.

This also applies to APRA-regulated super funds such as AustralianSuper:

Because SMSFs are Trusts, the ATO will automatically add “The Trustee for” to the name.

This is confirmed on the ATO website:
​https://www.ato.gov.au/businesses-and-organisations/trusts/trusts-registering-and-reporting-for-tax

No bank in Australia will ever record the name of an SMSF as The Trustee for XYZ SMSF.

The correct format for an SMSF bank account is

"SMSF TRUSTEE COMPANY NAME PTY LTD ATF SMSF NAME"
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Here is an example of a letter from Macquarie Bank that shows the correct naming format for an SMSF:

Provided the SMSF name (the words after "ATF") is shown on the bank statement and matches the SMSF name (the words after "The Trustee for") on Super Fund Lookup, the account is in the correct name and the bank statement/letter must be accepted by all APRA Regulated Super Funds.

It also needs to be pointed out that the bank account validation contained in the ATO's SMSF Verification Service (SVS) only validates the BSB and Account Number attached to the SMSF record with the ATO.
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The SVS does not validate the account name.