you are right with your thoughts.
"um" and "mag" are prefixes however the"um" verb could be place right before the vowel or right after the consonant.
Um and mag verbs can start either with a vowel or a consonant. Additionally, English loan verbs are mag verbs, but there is no clear-cut rule how to classify local Tagalog verbs in the um and mag paradigms.
lets take an example:
"mag" verbs
magcomputer (to use a computer) , magfile (to file), magsimba(to go to church), mag-ayos (to fix something in place), mag-utos (to command) mag-linis (to clean)
"um" verbs
sumigaw (to shout) , umeksena (to staged "eksena" is scene) , kumanta ( to sing) ,umayos (regulate/recondion) , umamin (to confess something)
Actually there is an example of acceptable use of <um>, <mag> and the nasal assimilation that are interchangeable in a sentence an example is sita,which means criticize.
Ang sama niyang magsita
Ang sama niyang sumita
Ang sama niyang manita
all of these could be (past, present,future) depending on how you will interpret these sentences. However all of them has the same meaning "how bad is he/she to criticized for "somethig specifically"...(for a matter of reasons)
acceptable in them...
um can be either infinitive or past tense but mag- has a past tense form and that is nag-