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This is correct, unless it says counter, it is not referring to physical counters on the card itself.
Additionally note the duration included on Giant Growth, while counters do not have a duration (or if they do they have a separate wording to remove them explicitly).
Fathom Mage checks for counters, and Giant Growth does not give counters. If it gave counters, it would say it gave counters.
When a term defined by the rules (e.g. "counter") is used, uses of that word only refer to that definition. "Attacked" does not refer to arbitrary offensive actions, "targetted" does not refer to objects affected by arbitrary spells, and "counter" does not refer to arbitrary things that modify power and toughness.
Giant Growth creates a continuous effect that modifies a creature's power and toughness.
Battlegrowth puts a counter that modifies a creature's power and toughness on a creature.
Fathom Mage's ability would only be triggered by the effect of the second spell.
A counter is a physical marker on a card that gives an effect while it's in play.
Giant Growth is a spell that when it resolves, gives a temporary effect until a specified time (i.e. end of turn).
Just because both give a boost to power and toughness in this case, doesn't mean that both are the same thing. As you say, both give stat improvements, but that does not imply that all stat improvements are of an identical and interchangeable type.