The spellcasting class feature is what really matters
There is no explicit definition of spellcaster, because it isn't an important term. When a rule requires precision it refers directly to the spellcasting class feature! That is, the '_____ spellcasting' feature which is usually named after the class' tradition, or the class itself (if the tradition varies).
For example, there isn't a single prerequisite that mentions 'spellcaster', but there are a few that mention 'spellcasting class feature'.
As another example, the rule for the Activate an Item activity, likewise only refers to the spellcasting class feature:
If an item lists “Cast a Spell” after “Activate,” the activation requires you to use the Cast a Spell activity to Activate the Item. This happens when the item replicates a spell. You must have a spellcasting class feature to Activate an Item with this activation component. If the item can be used for a specific spell, the action icon for that spell is provided. If it's an item like a staff, which can be used for many spells, the icon is omitted, and you must refer to each spell to determine which actions you must spend to Activate the Item to cast it.
That is the crux of your answer that sparked this questions: a monk with Ki Strike doesn't have a spellcasting class feature so it can't Activate an Item such as a spellheart.
Yet another example are the rules for Spellcasting Archetypes, which state:
Some archetypes grant you spellcasting abilities, albeit delayed compared to a character from a spellcasting class. In this book, the spellcasting archetypes are bard, cleric, druid, sorcerer, and wizard, but future books might introduce spellcasting archetypes that aren't multiclass archetypes. A spellcasting archetype allows you to use scrolls, staves, and wands in the same way that a member of a spellcasting class can, and the basic spellcasting feat counts as having a spellcasting class feature.
Spellcasting archetypes always grant the ability to cast cantrips in their dedication, and then they have a basic spellcasting feat, an expert spellcasting feat, and a master spellcasting feat.
For instance, a monk with the Druid Dedication feat can use scrolls, staves, and wands despite not having a spellcasting class feature, and if that monk also takes the Basic Druid Spellcasting feat then it counts as having a spellcasting class feature, so it would be able to Activate a spellheart too.
Cast a Spell does not make a spellcaster
Although there is no explicit definition of spellcaster, Chapter 7: Spells implies throughout that being a spellcaster is strongly related to a character's class (afterall, what matters is the spellcasting class feature) and this is especially evident in the section about Focus Spells:
Focus spells are a special type of spell attained directly from a branch of study, from a deity, or from another specific source. You can learn focus spells only through special class features or feats, rather than choosing them from a spell list. Furthermore, you cast focus spells using a special pool of Focus Points—you can’t prepare a focus spell in a spell slot or use your spell slots to cast focus spells; similarly, you can’t spend your Focus Points to cast spells that aren’t focus spells. Even some classes that don’t normally grant spellcasting, such as the champion and monk, can grant focus spells.
Spellcasters with Focus Spells
If you are a spellcaster, your focus spells are the same tradition of spell as the class that gave you the focus spell. A bard’s are occult, a cleric’s are divine, a druid’s are primal, a wizard’s are arcane, and a sorcerer’s are determined by their bloodline.
Non-Spellcasters with Focus Spells
If you get focus spells from a class or other source that doesn’t grant spellcasting ability (for example, if you’re a monk with the Ki Strike feat), the ability that gives you focus spells also provides your proficiency rank for spell attack rolls and spell DCs, as well as the magical tradition of your focus spells. You gain the ability to Cast a Spell and use any spellcasting actions necessary to cast your focus spells (see below). However, you don’t qualify for feats and other rules that require you to be a spellcaster.
This section explicitly categorizes bard, cleric, druid, wizard, and sorcerer as spellcasters, while monks (even with Ki Strike) are categorized as non-spellcasters. Champions are mentioned alongside monks for not granting spellcasting, so it's pretty clear that likewise they are not spellcasters (despite the fact that champions have focus spells as class features).
Monks (with Ki Strike) and champions can Cast a Spell because they have focus spells, yet they are not spellcasters, so being able to Cast a Spell is not sufficient to be a spellcaster.