Which zodiac animal does a birth date carry, and where is the boundary?
The Korean saju year turns at a solar term in early February, not on January 1, so a birth date in that window can carry the previous year's animal.
The animal is one character out of eight
The animal belongs to the year column. A chart in this system is four columns and eight characters, and the animal is the reading of one of them: the earthly branch that sits under the stem of the year.
Twelve branches run in a fixed order and each carries one animal — ja (子) rat, chuk (丑) ox, in (寅) tiger, myo (卯) rabbit, jin (辰) dragon, sa (巳) snake, o (午) horse, mi (未) goat, sin (申) monkey, yu (酉) rooster, sul (戌) dog, hae (亥) pig. The romanisation here is the Korean reading of each character.
The same twelve branches are used for the month, the day and the hour columns as well, so a full chart carries four of them, one per column. The one people mean by "my animal" is the year one, and the other three are not usually given animal names outside the chart.
Where the year turns
The year column does not turn on January 1. It turns at ipchun, the solar term at which the sun reaches 315 degrees of ecliptic longitude, and that instant moves from year to year rather than sitting on a fixed date.
In the calculation this site uses, that instant fell on February 3 or 4 in UTC for every year from 2020 to 2030. It is one instant rather than one date, so the calendar date it lands on depends on the clock it is read against, and the two can differ by a day.
That is what puts late January and the first days of February in question. A birth date before the boundary belongs to the previous year's branch and a birth date after it belongs to the new one. Where a boundary other than this one is applied to the same date, the answer can come out as a different animal; this page states which boundary this site applies, and does not try to settle which of them is correct.
Checking a date that falls in that window
The calculator on this site takes one birth date at a time and prints the four columns. For the year column it prints the two characters, a romanised reading and the element each of them carries. It does not print the animal name, so the branch character is what appears on screen and the list above is what converts it.
It also prints the solar term period the birth moment falls in and the start of that period in UTC. Those two rows are what make the answer checkable rather than something to take on trust: the boundary that decided the year column is on the screen next to the column it decided.
Within two minutes of a boundary the year and month columns are not produced at all. Independent calculations of the same solar term differ by around that much, so an input inside that band is left uncalculated rather than rounded to one side of it, and no animal follows from a column that was not produced.
What the answer settles and what it does not
An animal names a position in a cycle. Everyone born inside the same stretch of the calendar carries the same one, which is why the year branch on its own is the least individual part of a chart.
So this page answers the smaller question and answers it completely: which branch a date falls under, which boundary decided that, and how to check the boundary on the screen. What the tradition then reads from a chart needs the other three columns, and those need a time of birth as well as a date.
Open the chart calculator
The calculator prints the branch of the year column and the solar term that decided it. It does not print the animal name.
The calculator turns a date, a time and a time zone into four pillars and names every calendar rule it used.