Does this clause mean the contract is auto-renewed as he said?
Yes. However, your former employer's position on renewals would fail on grounds of (1) the doctrine of contra proferentem, and (2) the parties' subsequent conduct.
The second item  in this court decision (note that there are two consecutive items numbered "34") reflects that Uganda recognizes the doctrine of contra proferentem, and it is palpable from your description that the party who drafted the clause about "Consultancy period" was your former employer.
The clause on "Consultancy period" indicates a list of possible expiration/end/termination dates, two of them being May 30, 2019 and March 2020. Since you did not draft the contract, the aforementioned doctrine entitles you to the end date that favors your position.
Strictly speaking, your employment as of the time you resigned would be midway the 3-month period that began on September 1 and would end on November 30. However, the former employer's acceptance of your resignation indicates his agreement to void that part of the clause "Consultancy period". In other words, the parties' subsequent conduct implies that he forfeited his right to enforce the duration of that 3-month period.
Another aspect that would reinforce your argument of parties' subsequent conduct is whether or not the employer keeps paying you as if you hadn't resigned yet. If he does not, then it further evidences his understanding that you are no longer his employee, whence he cannot reasonably expect you to perform any additional work.
The former employer would have a valid claim of breach of contract only if the contract is project-based rather than --or in addition to-- duration-based. However, your description has no indication in that respect.