Antimagic has always prohibited the casting of any magic or magical abilities.
the 3.5 DMG note the following in Chapter 8 (Numbering mine, original is bullets):
- No supernatural ability, spell-like ability, or spell works in an
area of antimagic (but extraordinary abilities still work).
- Antimagic does not dispel magic; it suppresses it. Once a magical
effect is no longer affected by the antimagic (the antimagic
fades, the center of the effect moves away, and so on), the magic
returns. Spells that still have part of their duration left begin
functioning again, magic items are once again useful, and so
- Spell areas that include both an antimagic area and a normal
area, but are not centered in the antimagic area, still function in
the normal area. If the spell’s center is in the antimagic area,
then the spell is suppressed.
- Golems and other constructs, elementals, outsiders, and corporeal
undead, still function in an antimagic area (though the
antimagic area suppresses their spellcasting and their supernatural
and spell-like abilities normally). If such creatures are
summoned or conjured, however, see below.
- Summoned or conjured creatures of any type, as well as incorporeal
undead, wink out if they enter the area of an antimagic
effect. They reappear in the same spot once the field goes away.
- Magic items with continuous effects, such as a bag of holding, do
not function in the area of an antimagic effect, but their effects
are not canceled (so the contents of the bag are unavailable, but
neither spill out nor disappear forever).
- Two antimagic areas in the same place do not cancel each other
out, nor do they stack.
- Wall of force, prismatic wall, and prismatic sphere are not affected by
antimagic. Break enchantment, dispel magic, and greater dispel
magic spells do not dispel antimagic. Mordenkainen’s disjunction
has a 1% chance per caster level of destroying an antimagic field.
If the antimagic field survives the disjunction, no items within it
The first point is the most important; it absolutely precludes spells cast from within or upon those within. The heal spell thus cannot work, nor can the class abilities that heal instantly. That the spell is instant is of no matter.
The confusion seems to be based upon Point 2 and point 6. Since the spell fails, there's no spell to have continuing effect nor an instant effect.
Note that the lack of effect upon instant spells is upon ones already having gone into effect before the Antimagic Zone (AMZ) is encountered; such spells have literally already ended, and thus can't be affected. They still cannot be cast into, from, or through the AMZ.
Point 4, for reference is an explicit exemption for some types; point 5 is a clarification of point 4, noting what isn't exempted but could be confused for those that are.
Point 6 is a specific example of point 2.
Point 7 and 8 are both about ending the AMZ; point 8 also includes additional exempt spells, and does not include heal spells.