I've found two answers. One says its the determination of concentration of unknown alkali solution with the help of standard acid solution. The other says it as the determination of concentration of unknown acid solution with the help of standard alkali solution.
IUPAC's current terminology for the kinds of titrations is based on the following classification [1, p. 580]:
- nature of end-point detection (coulometric, potentiometric, amperometric, conductometric, thermoanalytical, optical, radiochemical titrations);
- nature of the reaction between analyte and reagent (acid-base, precipitation, complexometric, redox titrations);
- nature of the titrant (acidimetric, alkalimetric, iodometric, coulometric titrations).
So, alkalimetry is the term used to reflect the nature of the titrant, and the historical IUPAC's definition  is still correct, and shall be viewed as a type of acid-base titrimetric analysis:
2. Alkalimetry. Determination of a substance by titration with a base.
27.03. ALKALIMETRIC. An acid-base titration in which an acid is titrated with a standard solution of an alkali.
As for your confusion, it may be explained by the following footnote [2, p. 429]:
The term acidimetry has opposite meanings in different countries. For instance, in Britain and the U.S.A. it is used in both senses, i.e. determination of acid and determination with acid. In France acidimetry generally means measurement of acid, and this appears to be the original meaning of the term. However, all other usages of similar terms imply titration with, e.g. argentimetry; accordingly, to maintain consistency it is recommended that acidimetry means titration with. The same remarks apply to alkalimetry.
- Camões, M. F.; Christian, G. D.; Hibbert, D. B. Mass and Volume in Analytical Chemistry (IUPAC Technical Report). Pure Appl. Chem. 2018, 90 (3), 563–603. DOI: 10.1515/pac-2017-0410. (Free Access)
- Sandell, E. B.; West, T. S. Recommended nomenclature for titrimetric analysis. Pure Appl. Chem. 1969, 18 (3), 427–436. DOI: 10.1351/pac196918030427. (Free Access)
Practically, the right meaning is the meaning used and/or assumed in given context, circumstances, community. Use, what is more usual in your region, or perhaps better, replace it by more universal and less controversial/confusing acid-base volumetry.
I assume that the meaning of alkalimetry/acidimetry has the similar fate as the work sign convention schisma in the 1st law of thermodynamics for thermodynamic systems (work done on a system $\Delta U = Q + W$ versus work done by a system $\Delta U = Q - W$ ).
Some communities in some periods used terms acidi/alkalimetry for volumetric acid-base titration of acids/bases, as AChem suggests in his etymology analysis as measuring of acids/bases.
Some other communities in some periods used terms acidi/alkalimetry for titrations by acids/bases, like manganometry by permangate, complexo/chelatometry by EDTA etc. Manganometry or chelatonetry are not obviously volumetric analysis of manganese or EDTA contents.
Both notions have some logic in them, similarly as the mentioned opposite work sign conventions. So the right answer may be context dependent and it is always good -- as for TD work -- to explicitly write what you mean by the terms.
It is an obsolete term but nevertheless an analytical chemist should be familiar with it. Break the word into two: alkali + -metry. Measurement of an alkali (content). In other words, you use a standardized acid during titration, and the base concentration is unknown. The opposite is acidimetry.
Please note Wikipedia definitions of acidimetry and alkalimetry are incorrect. If you look at original books from 1890s or early 1900s even from respectable German authors, they define alkalimetry as defined in the answer. I pasted the screenshot of the unabridged version of the Oxford Eng. Dict. because it is not available to public. Maybe someone started the inconsistent definitions in some handbook and some people followed the same without giving a second thought.