I don't really get the logic behind this, yes, the term has more than one meaning/use in different contexts, just like virtually all words, and logging services have little to nothing to do with cameras. Furthermore, "tracking" clearly emphasizes the "always looking directly at the target" aspect, less so by "chase", in my opinion. Could you explain your reasoning on this?
But, "tracking" is more closely related to "A service logging the actions the user has done." Such as, "I created a tracking app." It may bring up mixed results. "Chase" on the other hand seems plausible enough. I'm just not sure.
Otherwise, in addition to suggested names above, "centered camera" springs to mind.