There are multiple different rules to choose from
The reason that you see so many contradictory takes is because the 5e rules provide a number of different ways to resolve areas of effect.
Narrative combat. If you use narrative combat, there is not even a floor plan to apply templates to. Instead, under Adjudicating Areas of Effect on p. 249, the DMG gives you a formula on how to calculate the number of affected targets. "This approach aims at simplicity instead of spatial precision. If you prefer more tactical nuance, consider using miniatures.". Narrative combat is actually the default, playing a grid is a "variant" rule (p. 192 PHB).
Just miniatures, No grid. The intro (p. 6 PHB) states you can use miniature figures "to represent each creature involved in a scene to help the players keep track of where everyone is.", but does not mandate a grid. That is, while a grid of 5-foot squares and minis is the most commonly used setup, you also can play without grid. In such cases, the templates can exactly fit the area descriptions in the spellcasting rules, with whatever scale for your minis you use. Partially being in the area of effect is being in the area of effect, so any amount of overlap by the template over a figure will mean the creature is affected.
5-foot square or hex grid, core rules. The core rules (DMG, p. 251) tell you to use an exact template of the shape given in the spellcasting section as your template, and then determine which squares are affected as follows1: Choose an intersection of squares or hexes as the point of origin of an area of effect, then follow its rules as normal. If an area of effect is circular and covers at least half a square, it affects that square.
The template method, an optional rule from Xanathar's Guide to Everything (XGE), p. 86, again uses templates in the exact shape from the spellcasting rules, but now any square that is overlapped by the template is affected. "If any part of as quare is under the template, that square is included in the area of effect. If a creature's miniature is in an affected square, that creature is in the area.", which makes it easier to decide if a square is affected or not, and increases the number of affected squares. It also recommends to use this method without a grid, and states "If you do
so, a creature is included in an area of effect if any part
of the miniature's base is overlapped by the template", the same as answer 2, with the slight difference that the base of the miniature is used, not the miniature.
The token method from XGE, p. 87, instructs you to "make areas of effect tactile and fun", by putting a 6-sided die or other token in each affected square of a square grid. I have never seen anyone actually use it, but for the layout of these dice, predefined patterns are provided. Cones now only work at orthogonally or diagonally, lines too, and circles of a given diameter here are depicted as squares as long on each side. This may explain why some of the templates you have seen depict cicular areas as squares (unless they are from 4e D&D, which also did that).
Readymade templates for 5-foot squares
Exact templates can be oriented at any angle, not just orthogonally or diagonally to a grid that you play on. However, if you use method number 3, it can save you a lot of time for circular areas of a given size (cylinder or sphere, where rotation angle orthogonally to the ground does not matter), or for squares and cones of a given size that are used orthogonally or diagonally to the grid, and for lines at some additional angles, to already prepare templates that show the affected squares, like here for Pathfinder. When you use these templates at these angles, you can directly see which square is affected. For example, here are the templates for affected squares on the ground for fireball in 5e, cast at various heights over ground. Note that because the rules differ between 5e and earlier editions, sometimes you run into templates online that show areas that would be incorrect under 5e rules, like the ones for a 20-foot circle from Pathfinder.
Spell areas of effect
These are all listed on p. 201, PHB, and any exact template that is correct would need to match them (and affected-square templates would need to match them after translating to squares following the rules). For example
A cone’s width at a given point along its length is equal to that point’s distance from the point of origin. A cone’s area of effect specifies its maximum length
That means at a distance of 20 feet, a cone would be 20 feet wide. This is not going to create a triangle with equal angles and sides if seen from above as a template, it is going to be more narrow.
The depiction of the areas shows that the end of the cone would be a flat circular cross-section. For Cones, Cubes and Lines, the caster can decide if the point of origin is included in the area, for cylinders and spheres it always is.
Placing the point of origin
The rules (PHB, p. 201) say:
Every area of effect has a point of origin, a location from which the spell’s energy erupts. The rules for each shape specify how you position its point of origin. Typically, a point of origin is a point in space, but some spells have an area whose origin is a creature or an object
When playing on a grid, for spells that have a normal point of origin, the core rules in 5e are clear
Choose an intersection of squares or hexes as the point of origin of an area of effect
However, there are spells that emanate from a creature, and in this situation the core rules are ambiguous. You could either pick a corner of the creatures grid cell, or the center of the grid cell. So to your question:
there are "grid circle" templates out there, but they place the point of origin on a cross in between 4 squares. Wouldn't a centerpoint of a single square be the point of origin on an AOE that originates from the "self"?
the answer is: both are valid ways to do it.
It also is interesting that the PHB rules (p. 201) say "The rules for each shape specify how you position its point of origin., but these rules only explain where to put the point of origin in the area of effect, not where it will be on the map. Spells like burning hands, cone of cold, and color spray instead explicitly tell us that the cone emanates from the casters hands, so that is where the point of origin must be.
So, to summarize, the reason that this is all confusing and you see a lot of variety is that the writers of 5e tried to be many things to many people, and did not stick to one way how to do this. Plus, there are wrong depictions and templates out there. You need to pick one way you will use, and consistently use that if you look to be clear and consistent.
1 This rule is incomplete, as it does not explain how to deal with other shapes that do not fully cover a square (or hex, for that matter). Most people assume they should use the same rule as for circular areas, and count squares that are overlapped more than half.