Those words are synonyms. They all mean the same, so you can use whichever you like better.
But your prepositions are wrong, and you forgot two synonyms:
- Auf der dritten Etage
- Im dritten Geschoss/Geschoß*
- Im dritten Stock
- Im dritten Stockwerk
- Im dritten Obergeschoss/Obergeschoß*
(*) The word Geschoss/Geschoß is one of the very rare examples, where the orthography of German German and Austrian German are different. The reason for this is the different pronunciation. The o in this word is short in German German, but long in Austrian German. And there is a rule that says: "After short vowels use ss, after long vowels use ß". And that's why the official standard-writing of this word is different in Germany and Austria. (I am from Austria, so I will write »Geschoß« here in this article.)
(Also note that in Swiss German, there is no ß. In Switzerland, every ß has to be replaced by ss.)
Etage is a little bit exalted.
The word Stock can also mean "stick" (like walking stick or drum stick), so if you want to be very clear, it is to better use Stockwerk instead of Stock.
Geschoß (and Obergeschoß) has, IMHO, a more technical connotation than the other words.
This Ngram shows you how frequently these synonyms are used:
Names of floors/storeys in German language
The level of the pavement is called Erdgeschoß. In elevators you will see either the letter »E« or the number »0«.
The next higher level is erster Stock or erste Etage. In elevators the number »1«. (Higher levels are zweiter Stock, dritter Stock etc.)
The first level below Erdgeschoß is called erstes Untergeschoß. In elevators the letter »U« or the number »-1«. If there are more levels below ground level, you will see »U1«, »U2« etc. or »-1«, »-2« etc.
But there are also some special names:
In buildings with only one level below street level, Keller is this lowest floor. But the word Keller is more bound to its usage than to its place in the building. A Keller is an area, where you can store goods. This means: A Keller exists only in buildings, where you have the need to store things for a longer period of time. You find a Keller in residential buildings, but it is very rare to have a Keller in office buildings.
I lived for many years in a multi-functional-building. There was (from bottom to top) an underground car park, a shopping mall, offices and apartments in this building. It had 4 levels below ground and 13 levels above ground. Residents needed storage space, so this building had to have a Keller, but in the lower 4 levels there was no space for it because of the underground car park. We had our Keller between the highest office level and the lowest residential level, on the 5th floor. This is very rare, but it shows that the meaning of the word Keller is more relevant to its purpose than to its position in a building.
In elevators, sometimes K for Keller is used, but this is rare. More often you just see U or -1 (if Keller = erstes Untergeschoß).
Just a synonym for Erdgeschoß. So in some houses you find a »P«-Button in the elevator, where you would expect »E« or »0«.
More and more people are moving into big cities, and most of them want to live near the center, but you can't build new buildings there because they've already been built. You can't make those buildings higher because laws often limit the height of buildings. In addition to that, many buildings are under monument protection, so you can't change their appearance. But you can convert attics into new apartments or lofts. This is called »Dachgeschoßausbau« (loft extension).
Those added levels may be called, for example, fünfter Stock (if the former last level was vierter Stock), but they very often are called »Dachgeschoß«. If there is only one Dachgeschoß in the building, then it is simply called »Dachgeschoß« (»D« in the elevator). If there is more than one, then you have »erstes Dachgeschoß«, »zweites Dachgeschoß« and so on (»D1«, »D2« etc.)
Special floor names in Vienna
The names of floors in Vienna are unique because during the time of the monarchy, a law prohibited building floors above the fourth floor. A fifth floor would be illegal.
But in 1910 Vienna was the 4th biggest Megacity in the world (only New York, London and Paris were bigger). About 2.1 million people lived there, and the city needed room for their residents. So, people built buildings with many more levels then just four. But to have the highest level be named »vierter Stock«, they needed to invent new names for the lower levels. So, in some old buildings in Vienna, you still find names for a building's levels (from bottom to top):
Typically the floor in Tiefparterre is one or two meters below street level, so you still can have windows at street level to let light and fresh air into the apartment.
Button in the Elevator: »T«
This is the level you would call Erdgeschoß or in English ground floor, although you have to step a few steps up to reach it.
Button in the Elevator: »P«
In "normal" counting this would be erster Stock, but not in old houses in Vienna.
Button in the Elevator: »H«
- Mezzanin (from Italian mezzo = half)
Literally this a half-level between two other levels, but in praxis it is just another level.
Button in the Elevator: »M«
- erster Stock
Did you count? We are already three floors above street level, and four levels above the lowest level for apartments, but still call it the first floor.
Button in the Elevator: »1«