Here's the Labour Party's constitution: http://labour.org.uk/wp-content/uploads/2019/04/Rule-Book-2019.pdf
The relevant section is Chapter 3, Clause III, Rule 3 - Procedural Rules for the Party Conference, voting. The rule begins:
Voting at Party conference on resolutions, reports, amendments, proposals and references back shall be by show of hands or, when the conditions laid down by the CAC require it, by card.
The CAC is the Conference Arrangements Committee, which is detailed in Clause II of the same document. At the start of the conference they present a report detailing Standing Orders of how the conference will run. The 2019 one is here. The relevant section is Appendix 5, Section 5.B.i - Voting on Resolutions:
Voting on resolutions, reports, proposals and references back shall be by show of hands. Where a show of hands is unclear a card vote can be called at the discretion of the Chair. A card vote is intended to resolve a position where a show of hands is not decisive, to establish the exact breakdown of votes when the majority is of procedural significance (eg two-thirds required) or on a challenge to the Chair.
So it seems to me that the rule for the 2019 conference on when a vote is required to be counted is whenever the chair decides to call one. In this instance, she didn't. So probably the letter of the rule was not breached. I would note however that the same section makes it clear that the possibility of a counted vote is intended to resolve problems where the show of hands is not decisive, or where there's a challenge to the chair. Both of those scenarios occurred; the vote was close enough for the chair to initially call it one way and her colleagues to disagree, and there were challenges from both the floor and subsequent speakers.