Interpretation of 'last day of rental period' under periodic tenancy

My openrent contract states:
11.2 If the tenancy has lapsed into a periodic tenancy it may be terminated by:
• the Tenant giving written notice of at least one month and expiring on the last day of a period of the tenancy.
My tenant has just given 4 weeks notice (16.08.20). Rent is paid on 5th of every Month, so I believe the period of the tenancy is from the 4th to 5th of each month now that it is a periodic tenancy.
The tenant has been told (by the agents acting for the LL to where they are moving) that their rental obligations finish on 16.09.20 - 4 weeks from when they gave notice. My interpretation of the clause above is that their rental obligations should be carried to the 4th Oct (04.10.20). Any help with this would be welcome in the effort to avoid a possible dispute with the tenant. Thank you in anticipation of your comments.

My understanding from reading the housing act a while ago is if a tenancy started, let’s say, on the 5th of a month then after the fixed period monthly periodic tenancies would run from each 5th of the month until the 4th of the following month. I recall from the act that a full periods notice must be given, so that notice given on 16th in your case should really expire on 4th of the month after next (about 6 weeks)-others may correct me if my understanding is wrong- however my view is it is better to work with the tenant to achieve a smooth transition than to argue the toss over who had the right legal interpretation. What would you do if, for example, your tenant did not pay a full or perhaps any last month rent? It would only come out of the deposit and you’d be exposed if any damage had been caused. Better in my view to agree a convenient leave date and both work towards it, perhaps including you showing new prospective tenants around to minimise your unlet period. Hope this helps.

Thanks for your comment Steve. You seem to agree with my understanding of the periodic term and notice to quit. And you are quite right about avoiding strife with tenants (if possible) too!