I’ve read some sources that say a syllable is “one beat” but I don’t understand that. Wouldn’t it depend on the tempo of the pulse. I.e, if a tempo is 60bpm can’t you fit different numbers of syllables between each beat depending on how fast you are uttering? I’ve also read that a typical speaking rate for English is 4 syllables per second which means a syllable could be a half a second (eighth note), or even a half of a half of a second (sixteenth note), is this correct?
A syllable is an organizational abstraction, a grouping of consonants around a vowel (with various putative exceptions). Beyond that, it doesn't make any sense to try to "define" a syllable, but it does make sense to ask how or if one can detect one.
There are various Western cultural traditions related to writing that are used to determine that some substring of an utterance is a syllable. English speakers are taught from an early age to speak slowly and "sound out" a word, so that a word like "catastrophic" is performed at an abnormally slow rate, and you can consider each production chunk to be a syllable. The vowel bears the brunt of this prolongation: you can then consider each syllable to be "a beat", except that "beat" is not a linguistic term, it is a musical one taken into linguistics.
In fact, "beat" is the wrong concept for syllables, a "beat" more closely corresponds to the metrical foot, which is a grouping of two (or three) syllables.