The lunar stations are laid out so that the Moon resides about 1 day in each.
The Moon takes an average of 27.32 days between passes of the same star (sidereal month) or ecliptic longitude (tropical month),
so some traditions have 28 stations and some have 27.
Station definitions can be either sidereal or tropical, and their widths can either be equal or vary with the spacing between stars.
In any case, the associated calendar dates are when the Sun passes the reference points for those stations.
The spreadsheet NOAA_Solar_Calculations_year can help you determine the Sun's ecliptic longitude for a given date or vice versa.
If you set the year and time of day in column B, the date is in column D and the Sun's longitude is in column P.
For example, at 0:00 UT on 2021-01-27, the Sun is at longitude λ☉ = 307.3°.
The dates in the question reflect two different station reckoning methods, which produce different results.
The April 23-May 10 and April 20-22 dates fit a variable-width sidereal scheme.
If you put a manzil start date from this article into Stellarium and disable the atmosphere [A], the Sun appears near the corresponding star,
e.g. Aldebaran (α Tau) on May 30 (Al-Dabaran)
or Shaula (λ Sco) on Dec 1 (Al Shaulah).
You can reproduce those dates by finding when the Sun has the same ecliptic longitudes as those stars.
The longitudes above are relative to the J2000 equinox; for other years, add 0.014°/year for precession.
To apply this method to the 2021-01-27 example, add 0.3° precession, putting β Cap at 304.3° and μ Aqr at 313.4°.
The Sun at λ☉ = 307.3° is 32% of the way through station 22.
The table from the Wikipedia article describes a different manzil scheme.
The "period" column suggests an equal duration of 13 days, while the "starting degree" column shows an equal width of 12.86°.
Since the Earth's angular speed around the Sun varies depending on the time of year, stations can have equal duration or equal width but not both.
Let's choose equal width.
The table refers to a sidereal zodiac but doesn't state its offset from the tropical zodiac.
WikiBlame finds that it was added to the article in 2016.
For each station, I subtracted the longitude in the table from the solar longitude at 0:00 UT on the given date, averaged over years 2010-2021, and found a mean offset of 13.2° relative to the J2000 equinox.
For year Y, let's say station n starts at ecliptic longitude
$$ \lambda_n = 12.86^\circ (n - 1) + 0.014^\circ (Y - 2000) + 13.2^\circ $$
Returning to the 2021-01-27 example,
λ23 = 296.4°
and λ24 = 309.2°,
so the Sun at λ☉ = 307.3°
is 85% of the way through station 23.
Which method you prefer is a matter of opinion.