δεσμός - "Bond"
The first word - δεσμῶν - is a particular form (genitive plural) of δεσμός (desmos). The root of this word is the verb δεω (deō), which means to tie or bind.
The KJV translates δεσμός as "bond" or "band" in all but 2 of its 20 occurrences in the New Testament. The ESV (which I believe you quote) is a little freer in its translation and only uses "bond" on 4 instances, favoring "imprisonment" (9 times), "chains" (4 times), and even "bound with chains" (2 Timothy 2:9).
ἅλυσις - "Chain"
The second word - ἅλυσιν - is a form of ἅλυσις (alusis), which is the chain material itself. It is less common than δεσμός in the Bible, appearing 12 times in the NT+LXX vs. 68 times for δεσμός. Both the KJV and the ESV translate ἅλυσις without exception as "chain".
In classical Greek, The word appears in the Tactica of Arrian of Nicomedia to describe a link in chain armor1, as well as in Euripides' Eurestes:
Oh! to reach that rock which hangs suspended midway between earth and heaven, that fragment from Olympus, which swings on chains of gold.2
So I think your supposition is correct: δεσμός represents a "bond" in general, and ἅλυσις represents a particular type of bond ("chain"). Because bonds seem almost always to be with chains, some versions use "bond" and "chain" interchangeably when translating δεσμός. One can be bound with chains, but probably not chained with bonds:
μιᾷ γὰρ ἁλύσει σκότους πάντες ἐδέθησαν
For with one chain of darkness all were bound (Wisdom 17:16)
I do not read much into either word being in the singular or plural. One could be held with a single bond or chain or with multiple bonds and chains, I think:
And ought not this woman ... be loosed from this bond? (δεσμός - singular) - Luke 13:16
But he would break the bonds ... (δεσμὰ - plural) - Luke 8:29
Because of the hope of Israel I am wearing this chain (ἅλυσις - singular) - Acts 28:20
Peter was sleeping between two soldiers, bound with two chains (ἁλύσεις - plural) - Acts 12:6
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