Happily

adv.joyful, hopefuleveryday

Meaning 1

With visible pleasure or a settled feeling of contentment.

Reach for it when describing the joyful manner in which someone acts or lives

The cousins chatted happily while sorting the old photographs.

playful
merrily
the action has a light, festive rhythm
not when Merrily sounds storybook-like in sober reporting or workplace prose.
everyday
cheerfully
good spirits are outwardly bright and encouraging
not when Cheerful behavior can be put on even when the person is not inwardly happy.
warm
contentedly
quiet satisfaction matters more than excitement
not when It is too placid for laughter, celebration, or lively pleasure.

With ready and sincere willingness.

Reach for it when offering help or agreement in a friendly, positive way

I would happily show the next group around the garden.

everyday
gladly
the person welcomes the request without hesitation
not when Gladly expresses consent, not the broader happiness surrounding an action.
professional
willingly
free consent needs to be explicit
not when Willingness may be dutiful and neutral rather than enthusiastic.

By fortunate circumstance or with a welcome outcome.

Reach for it when introducing good news that might otherwise have gone differently

Happily, the rain held off until every roof was covered.

formal
fortunately
luck or circumstance produced the beneficial result
not when Fortunately cannot describe the joyful way a person performed an action.
literary
mercifully
relief from an unpleasant possibility colors the good result
not when It adds an avoided hardship that happily does not require.

The definition and the example on this page are written here. The dictionaries above ground which meaning is being described, not the wording.