Why the results all agree and all stop
Birthday pages are usually built from a table of sign date ranges and a stock paragraph for each sign. A date goes in, a sign comes out, and the paragraph is the same for everyone born across a month.
That method is accurate as far as it goes, and it cannot go any further. A date range cannot report a degree, because a degree depends on the year and the hour, and it cannot report a decan, because a ten degree band is finer than the tables are built to resolve. Which is why a hundred pages about 3 June say the same thing. They agree by construction rather than by checking anything.
The degree, which the date really does fix
Taken at noon in every year from 1930 to 2035, a 3 June Sun has stood between about twelve and thirteen and a half degrees of Gemini. That is a window of roughly a degree and a half across more than a century, which is narrow enough to be worth quoting.
The narrowness comes from the leap year rule. The quarter day of surplus in the tropical year builds up for three years and is cleared by the leap day, so the calendar and the orbit stay in step and a date holds its place on the ecliptic. Within one year the hour of birth moves the Sun by less than a degree from midnight to midnight.
The face, which distinguishes one Gemini from another
Twelve degrees of Gemini falls in the middle decan, which runs from ten to twenty. The Chaldean sequence of Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter, counted from zero degrees of Aries, gives that face to Mars. The triplicity scheme gives it to Libra, and so to Venus.
This is the layer that separates a 3 June Gemini from a 23 May one, and it is the layer stock paragraphs cannot reach. Tradition reads the Mars version as the sign given an edge and the Libra version as the sign turned towards company. Both are interpretation and both depend on the scheme, which is why an honest description names it.
And the parts nobody can read off a date
The Moon changes sign every two and a half days or so, so a 3 June birth can have it in one of two signs depending on the hour, and across years it has been in all twelve. The rising sign depends on the birth time and the birth latitude and can be any of the twelve on this date.
The widget above works from a computed position rather than a lookup table. Enter a year, a time and a place and it returns the Sun's exact degree and minute, the decan, and how far the position sits from the nearest sign boundary, which on this date is a comfortable distance in both directions.
None of that predicts anything. It states where the Sun was, names the band that position belongs to, and marks the traditional readings of that band as readings.
Questions people ask about this
Is a 3 June birthday always Gemini?
At noon it has been Gemini in every year from 1930 to 2035. The Sun is roughly two weeks past the Taurus boundary and three weeks short of Cancer, so nothing in the span puts it in doubt.
How far into Gemini is 3 June?
About twelve to thirteen and a half degrees, so a little under halfway through the sign. The widget above gives a single figure once it has a year, a time and a birthplace.
Which decan does that put me in?
The second decan of Gemini, from ten to twenty degrees. Its ruler is Mars in the Chaldean order and Venus in the triplicity scheme, through Libra.
Why does the degree matter at all?
It determines which aspects the Sun makes to other planets, which decan applies, and how a progressed chart advances. The sign label alone carries none of that information.
Do all birthday pages really say the same thing?
Most are generated the same way, from a range table and a per sign paragraph, so they agree by construction rather than by verification. The disagreements tend to appear only in the adjectives.
What is worth looking up next?
The Moon sign and the rising sign, since the date settles neither. A birth time is needed for both, and a birthplace for the rising sign and the houses.