I'd like to have a more precise answer to when a spell fizzles. The rule seems to have changed to when not all targets are invalid.
For example: If an opponent casts Essence Capture to counter my creature spell and add a +1/+1 counter to a creature he controls and I destroy his creature in response, the second target is now invalid and his creature won't get the +1/+1 counter. Is my creature spell countered or does Essence Capture fizzle?
The part of the "fizzling" rules that your question is about haven't changed recently. The rule quoted in the page you linked to, 608.2b, says in part
If all its targets, for every instance of the word "target," are now illegal, the spell or ability doesn't resolve.
In the example you gave, one of the targets is still legal, so the spell still resolves.
The part of the rule that changed recently is that it used to say that the spell would be "countered", but now it says that the spell "doesn't resolve".
The rule on "Fizzling" spells has changed recently, but not in the way your question mentions. Before when a spell had no legal targets it was countered by the game itself, now the spell just doesn't resolve and is put into the graveyard (or ceases to exist if it is not a card). For the most part there's very little difference between the two, the only one I can really think of at the moment is the new rules will not trigger Multani's Presence since the spell you controlled was not countered.
Under almost all conditions, a spell that has targets will not fizzle (now defined as being put into the graveyard without resolving) unless all of the targets become illegal. A spell that still has some legal targets will do what it can to every still legal target. Here's the rule, taken from the comprehensive rules (emphasis mine):
608.2b If the spell or ability specifies targets, it checks whether the targets are still legal. A target that's no longer in the zone it was in when it was targeted is illegal. Other changes to the game state may cause a target to no longer be legal; for example, its characteristics may have changed or an effect may have changed the text of the spell. If the source of an ability has left the zone it was in, its last known information is used during this process. If all its targets, for every instance of the word "target," are now illegal, the spell or ability doesn't resolve. It's removed from the stack and, if it's a spell, put into its owner's graveyard. Otherwise, the spell or ability will resolve normally. Illegal targets, if any, won't be affected by parts of a resolving spell's effect for which they're illegal. Other parts of the effect for which those targets are not illegal may still affect them. If the spell or ability creates any continuous effects that affect game rules (see rule 613.10), those effects don't apply to illegal targets. If part of the effect requires information about an illegal target, it fails to determine any such information. Any part of the effect that requires that information won't happen.
Because your example, Essence Capture, still has a legal target, your creature spell on the stack, it will not fizzle, it will resolve, doing everything it can, in this case just the spell countering, then be put into the graveyard as normal.
There are exceptions to this, spells or abilities that specify all targets must be legal when the spell resolves, or allow the ability to resolve if there are no longer legal targets. For example Gilded Drake who's ability resolves even if the target is no longer legal (thus sacrificing itself).
