What transposing actually changes
Every chord moves by the same interval and nothing else moves at all. The melody keeps its exact shape — the same distances between the same notes — and comes out higher or lower, which is what you want when a song was recorded by a voice that is not yours.
The number on the stepper is the distance from the key the sheet was written in, not a new key name. Two up means two up from wherever this song started, on this song and on every other one, which is the version of the instruction you can use without first working out what key you are in.
Sharps or flats
A semitone above A can be written Bb or A#. Same shape, same sound, same everything except what is printed above the syllable — so the choice is yours rather than the arithmetic's, and the two buttons above make it. Keys with flats in them read better with flats; keys with sharps read better with sharps, and mixing the two on one page is legible but distracting.
A capo is a different question
Transposing changes what the room hears. A capo changes only what your hand holds — clamp it on and the shapes you finger are lower than what comes out, so the sound stays exactly where it was. If what you actually want is easier shapes for the same key, that is the capo calculator, not this page. The difference is worth ten minutes: what your fingers play and what the song is in.
Why ChordPro survives this and a column layout barely does
If your sheet is ChordPro — chords in square brackets, inside the line — transposition is exact. The chord is attached to a syllable, so renaming it moves nothing.
If your sheet has the chords on a line above the words, the alignment is doing the work, and a longer name has to fit in the space the shorter one had. This page keeps every chord in its own column and tells you when one could not stay there: A becoming Bb with a single space in front of the next chord has nowhere to go. That is a real limit of the layout rather than a bug here, and it is most of the argument for keeping your songs in ChordPro instead — a converter for that is one page over.
Nothing leaves your browser
The transposition happens on your own machine, in the page you are looking at. There is no upload, no account and no copy kept anywhere — which matters if what you are pasting is a song nobody has heard yet.
When a stepper stops being enough
This page moves one sheet, once. What it cannot do is remember: that this song sits two down for your voice, that the capo lives on the second fret for it, that the singer reads it in Italian while you read the letters. That is what Strumfolio is for — the same transposition as here, kept per song, on every device you open it on, and readable with no signal once the page has been opened.