Sorry for answering my own question, but after hearing from Zaccagnini I found a reference to the variant of Abel's summation formula mentioned in the early version of the OP, where $f^\prime$ was assumed to exist and be continuous everywhere in $[\lambda_1, \infty[$: The result (with some extra) appears as Theorem 6 in Chapter VII of

K. Chandrasekharan, Introduction to Analytic Number Theory, Grundlehren math. Wiss. 148, Springer, 1968.

As I don't have much hope that a variant of the formula in the lines of the latest version of the OP has never appeared in print (in spite of the fact that it's straightforwardly generalized from the variant in Chandrasekharan's book, once you know you have a suitable version of the fundamental theorem of calculus for the Henstock-Kurzweil integral), I'm going to accept this as an answer.

Sorry for answering my own question, but after hearing from Zaccagnini I found a reference to the variant of Abel's summation formula mentioned in the early version of the OP, where $f^\prime$ was assumed to exist and be continuous everywhere in $[\lambda_1, \infty[$: The result (with some extra) appears as Theorem 6 in Chapter VII of

K. Chandrasekharan, Introduction to Analytic Number Theory, Grundlehren math. Wiss. 148, Springer, 1968.

As I don't have much hope that a variant of the formula in the lines of the latest version of the OP has ever appeared in print (in spite of the fact that it's straightforwardly generalized from the variant in Chandrasekharan's book, once you know you have a suitable version of the fundamental theorem of calculus for the Henstock-Kurzweil integral), I'm going to accept this as an answer.

Sorry for answering my own question, but after hearing from Zaccagnini I found a reference to the variant of Abel's summation formula mentioned in the early version of the OP, where $f^\prime$ was assumed to exist and be continuous everywhere in $[\lambda_1, \infty[$: The result (with some extra) appears as Theorem 6 in Chapter VII of

K. Chandrasekharan, Introduction to Analytic Number Theory, Grundlehren math. Wiss. 148, Springer, 1968.

As I don't have much hope that a variant of the formula in the lines of the latest version of the OP has never appeared in print (in spite of the fact that it's straightforwardly generalized from the variant in Chandrasekharan's book), I'm going to accept this as an answer.

Sorry for answering my own question, but after hearing from Zaccagnini I found a reference to the variant of Abel's summation formula mentioned in the early version of the OP, where $f^\prime$ was assumed to exist and be continuous everywhere in $[\lambda_1, \infty[$: The result (with some extra) appears as Theorem 6 in Chapter VII of

K. Chandrasekharan, Introduction to Analytic Number Theory, Grundlehren math. Wiss. 148, Springer, 1968.

As I don't have much hope that a variant of the formula in the lines of the latest version of the OP has never appeared in print (in spite of the fact that it's straightforwardly generalized from the variant in Chandrasekharan's book, once you know you have a suitable version of the fundamental theorem of calculus for the Henstock-Kurzweil integral), I'm going to accept this as an answer.