Very good translation, Lucy.  I'm sending more lyrics.  No reindeer.  This song definitely refers to horses.  Note below "bells on bob tails ring".  Your translation is great, but the lyrics in English you have are wrong, I believe.  Compare the two. This Christmas song, I understand, started its life as a Thanksgiving Day song.  Right mood, wrong holiday. 

Dashing through the snow
In a one horse open sleigh
O'er the fields we go
Laughing all the way
Bells on bob tails ring
Making spirits bright
What fun it is to laugh and sing
A sleighing song tonight

Oh, jingle bells, jingle bells
Jingle all the way
Oh, what fun it is to ride
In a one horse open sleigh
Jingle bells, jingle bells
Jingle all the way
Oh, what fun it is to ride
In a one horse open sleigh

A day or two ago
I thought I'd take a ride
And soon Miss Fanny Bright
Was seated by my side
The horse was lean and lank
Misfortune seemed his lot
We got into a drifted bank
And then we got upsot

Oh, jingle bells, jingle bells
Jingle all the way
Oh, what fun it is to ride
In a one horse open sleigh
Jingle bells, jingle bells
Jingle all the way
Oh, what fun it is to ride
In a one horse open sleigh yeah

Jingle bells, jingle bells
Jingle all the way
Oh, what fun it is to ride
In a one horse open sleigh
Jingle bells, jingle bells
Jingle all the way
Oh, what fun it is to ride
In a one horse open sleigh