Some offences may be prosecuted by, in addition to the CPS, other public bodies, like local authorities or transportation authorities. Even while these are public and not private bodies, would one still refer to these as a “private prosecution”?
I would say that these are public prosecutions.
There are not that many differences in principle between public and private prosecutions. Among those differences are that there are some variations in the procedural rules, and that public prosecutors are subject to judicial review for the proper exercise of their powers.
For example, the Criminal Procedure Rules, Part 7 ("Starting a prosecution in a Magistrates' Court") says in paragraph 6 that "an application for the issue of a summons or warrant must concisely outline the grounds for asserting that the defendant has committed the alleged offence or offences", along with other conditions. But this does not apply when the prosecutor is a "public authority" covered by section 17(6) of the Prosecution of Offences Act 1985. (This is the Act which created the Crown Prosecution Service as the principal public prosecuting authority in England and Wales.) One example of a public authority is the CPS, but so is a local authority as in the question, or "any other authority or body whose members are appointed by Her Majesty or by any Minister of the Crown or government department or whose revenues consist wholly or mainly of money provided by Parliament". So a local authority is on the same footing as the CPS in this regard.
Judicial review is available in circumstances where prosecutors might have abused their discretion (whether on grounds of human rights violations, unreasonableness, etc.). These are most familiar for the CPS, as there are many cases of the form Somebody v DPP (the Director of Public Prosecutions who heads the CPS). But the same can be seen in R (Bermingham) v Director of the Serious Fraud Office  EWHC 200 (Admin) where it was the SFO's decision not to investigate and prosecute which was being reviewed. There are also cases of local authority prosecutions being reviewed in the same way.
This is not the case with an ordinary private prosecution, as judicial review is only available for decisions made by public bodies. In 2019, Boris Johnson mounted a successful JR against the decision of a District Judge to issue a summons in a private prosecution against him; this was not a JR of the prosecutor's decision. (In Scotland, judicial review can operate against a private entity, but there are also hardly any private prosecutions in Scotland.) JR would also be available for any local authority prosecution, not just one mounted under a specific authorizing statute such as the Protection from Eviction Act 1977 cited in comments. What matters is that it is a decision made by a public body, an arm of the state.
For all these reasons, I would say that a local authority prosecution more closely resembles a "public prosecution" done by some other government body, than a "private prosecution" done by a random individual.
Prosecution is always by the government, who in the UK (per the abbreviation CPS for Crown Prosecution Service) is referred to as the Crown. In other jurisdictions they may be referred to as the State or the People.
The full title of the case will be “Name of Authority v X” which can be spoken as such (noting that v is “against”, not “verses”) or as “The Crown against …”.
If the authority brings a civil case, say for breach of contract, then that is not a prosecution. The case name would be in a similar form but they wouldn’t be referred to as the Crown. Similarly, if a private person brings a private prosecution, the same naming conventions apply, however, these are almost impossible to bring and, if there are any prospects of success, are invariably taken over by the Crown.