Well, I wind my own voice coils, so deal with it all the time.
There is some truth to that, but wire weight is only part of the full picture.
For the same speaker, same specs, comparing two same diameter coils (that's a given) and winding length , the 4 ohms one will use somewhat thicker wire than the 8 ohms one, so will be slightly heavier .
How much? ..... think some 20% heavier.
Same rough ratio applies between 8 and 16 ohms.
But, putting it into numbers, the coils will weigh around 2 or 2.5 grams, while a typical 12"guitar cone weighs around 12 grams (American sound) to 15 grams (British sound) to 18-25 grams (PA/Jazz speakers), you also have around 0.5 to 1 gram adhesive plus most of the spider (the far end does not move), also tinsel wires , eyelets and solder, and the dustcap.
Considering all, a 20% variation in a 2.5 gram part does not change the game by much.
See it by yourself: some Thiele Small parameters include Mms , Moving Mass, which is the grand total of what I've been naming above, in a clasic Jensen C12N it's around 25 grams, now you know why
Sadly Eminence has been dumbing down its datasheets and does not post that any more.
Celestion? .... they were never very open publishing speaker Tech data.
So of course a small change in weight will change resonant frequency somewhat, but it's not an *electrical* problem, just a weight/mechanical one.
Almost forgot: on *some* brands, for 16 ohms speakers they would need to go to a very thin, harder to wind wire, so often they cheat a little, us a not-so-thin wire, and wind it a little longer than otherwise needed (you'll see it reflected in a higher X-max spec), so *that* longer coil will be somewhat heavier than expected.
Speaker design is a tight compromise so sometimes tough decisions have to be made.
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