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I suspect up until now this was treated as just Warm-Up Boss (which mentions a boss immediately after the tutorial), but I think there might be enough examples for this as a subtrope. Often a boss-as-tutorial (not to be confused with Training Boss, which seems more like a scaled-up Training Dummy and often appears later alongside new mechanics). May be a Final Boss Preview, particularly if the game then flashes back to How We Got Here.
If this does go to TLP, might I suggest: In Medias Boss
Edited by UnsungYou mean an intro boss, right? They can be either Warm-Up Boss or Hopeless Boss Fight.
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Exactly What It Says on the Tin. Usually a Warm-Up Boss for obvious reasons. Example: Chaos 0 on Sonic Adventure.