There is certainly a difference I would say.
A stone seeing this was touched with pity, and, wishing to help the
cock, he laid himself across the stream.
Who still remembered how much his countrymen were indebted to Columbus
; and was touched with pity for the man who had performed such great
In the above example the anthropomorphised stone has been so 'influenced' by what he saw (which resulted in pity) that he has taken up and gone and made a stepping stone of himself in the stream. In the second sentence, the cock is 'influenced' with pity, when he recollects the 'great actions' performed by Columbus.
In both cases, pity is present, but it is not the underlying motivator. It was the stone's sight, 'seeing this' that catalysed the pity within him. It was the cocks remembering, 'still remembered', that resulted in the feeling of pity.
Then the knight is in a predicament, as he thinks and ponders over the
question: whether to present to her the head she asks him to cut off,
or whether he shall allow himself to be touched by pity for him.
The apostle, touched by pity in the man's voice, and the plight of his
handicap, looked straight at him
OED breaks the uses of the preposition by down into seven categories. The 5th is the category in which the word is being used here: Medium, means, agency.
Pity is the underlying force at work here. As such we focus more on the pity itself. Pity is the agency at work, the medium motivating the action. Is the knight going to be 'touched by pity' and therefore have his actions influenced?..
The apostle was driven 'by pity' in and of itself, to look 'straight at' the man.
I would say in the second two examples (by pity) our attention is placed firmly on the motivational force of the pity. Whereas in the first examples (with pity) our focus is more on the event that precipitated the pity, and on the general sense of pity that now consumes the two subjects.