Guides
The same positions, twelve different cuts
The planets are where they are. What changes between systems is where the twelve houses fall, which is why the system is written on your certificate.
What a house system decides, and what it does not
When two charts of the same moment disagree, it is almost never because the planets are in different places. Planetary positions come from the ephemeris and are not a matter of opinion: the Sun was at 11 degrees of Leo or it was not. What changes is where the lines dividing the wheel into twelve sectors are drawn.
Think of a plot of land: the ground is the same and the boundaries can be laid out in several ways. Each house system is a different way of laying them out. None of them is the true one, so the point is not to pick the correct system but to know which one was used.
Placidus, the one you will get by default
Placidus divides time rather than space. It takes the arc each degree of the zodiac needs to climb from the horizon to the midheaven and cuts it in three. It has been the most widespread system in modern Western astrology since the seventeenth century, and it is what most software opens with.
It has a known physical limit: above the Arctic Circle some degrees never rise at all, so the division becomes impossible or meaningless. For a birth that far north, another option is worth considering.
Whole Sign, the oldest and the simplest
In whole signs the ascendant's sign is the entire first house, 0 to 30 degrees, and every sign after it is a house. There are no intermediate cusps, no intercepted signs, and no latitude that breaks it.
It is the system of Hellenistic astrology, the one in use through most of the first millennium, and it has come back strongly. Comparing your chart in Placidus and in whole signs, some planets change house: that is exactly what should happen, not anyone's mistake.
Koch, Regiomontanus, Equal
Koch divides using the latitude of the birthplace as its reference and became popular in German-speaking astrology. Regiomontanus divides the celestial equator and was the medieval standard, the one horary charts were cast with for centuries. Equal houses draw exactly 30 degrees from the ascendant: it keeps the ascendant and gives up on the midheaven being the cusp of the tenth.
All of them answer the same question with different geometry. None is an improved version of another.
Why we write it down
A printed chart that does not declare its house system is a drawing nobody can check. Someone reading it in ten years will not know which convention they are looking at, and a different house can change the reading of a planet entirely.
So the certificate that ships with every order states the system used, along with the resolved coordinates and the time zone. It is not packaging: it is the only way the piece stays verifiable once we are no longer standing next to it explaining.
Which one should I choose?
With no preference, Placidus: it is what most readers will expect. For a birth at high latitude, or if you follow someone working in the Hellenistic tradition, whole signs. What the choice does not change is the Sun sign or any planetary position.
Does my sign change?
No. The Sun sign depends on where the Sun was, and no house system touches that. What can change is which house that Sun falls in, which is a different question.
Can I order the same chart in two systems?
Yes, they are two separate pieces. Each arrives with its own certificate declaring the system it was drawn with, so they can be compared without confusion.