Harvard Dict. of Music:
Piano. ... I. General. ... The modern piano is equipped with two or (especially in America) three pedals, the damper or "loud" pedal at the right, the una corda or "soft" pedal at the left, and (if present) the sostenuto pedal in the center. On being depressed, the damper pedal raises all the dampers, allowing all the strings to vibrate ...
Sustaining pedal. The sostenuto pedal (middle pedal) of the *piano. The term is sometimes used for the damper pedal (right pedal).
Random House unabridged:
damper pedal - [Music.] A pedal on a piano that when depressed with the foot raises the dampers and permits the strings to vibrate and sustain the tone. Also called loud pedal.
Webster's 3rd unabridged:
damper pedal - the pedal that controls the set of dampers on a piano — called also loud pedal, sustaining pedal
sostenuto pedal - a selective damper pedal on a piano that enables the performer to sustain selected tones or chords
sustaining pedal - 1: DAMPER PEDAL. 2: SOSTENUTO PEDAL.
w, you need to cite a source defining which pedal the Dämpferpedal is in the first place: the loud one, or the soft one. I checked DWDS but it didn't have it. Since you're the one who wants to know, you should provide that information.
Damper pedal definitely exists in English, but I'm not sure if it's a false friend or not in this case. I looked it up just to refresh my memory, and it's evidently actually another term for the loud or sustaining pedal (the one on the right). That makes sense, because it raises the dampers, allowing the sound to be sustained, but it's confusing because it doesn't dampen the sound the way the soft pedal does. But usage might be somewhat different in German, because we have a different word, mute, for the thing you use with other instruments, like strings or brass, to soften the sound rather than stopping it entirely.
I've never heard muffler pedal before, and it's not listed in any of my dictionaries. I assume it would mean the other pedal, what I think of as the soft pedal (the one on the left). Unless it's BE, it sounds a bit childish, or as if someone was just guessing at the word and didn't know the right one. I wouldn't have been able to come up with either una corda pedal (left) or sostenuto pedal (middle) myself without looking in the dictionary.