From new moon to full moon and back
Around new moon, little of the illuminated side faces us; at full moon, we see an almost fully lit disc. The quarters lie between these stages. At first quarter, the visible illuminated fraction is increasing; at last quarter, it is decreasing. A half-lit disc is called a quarter because of its place in the phase cycle, not because only a quarter of the visible disc is bright.
Waxing and waning describe a direction
Waxing means that the illuminated portion we see is growing; waning means it is shrinking. Between a quarter and a full moon, more than half the disc is lit. Between new moon and a quarter, we see a crescent. The transition is continuous: familiar phase names mark points in an ongoing movement, rather than separate shapes that suddenly replace one another.
A calendar is not a local weather forecast
A calculated phase moment needs a time zone to be displayed as local clock time. AstroNivo uses Europe/Amsterdam. An illuminated percentage does not show whether your sky is clear or the moon is above your horizon. The orientation of a crescent can also differ with observing location. When planning an observation, check more than the phase name alone.
Symbolic practices are a personal choice
Some readers use a new moon to write down an intention and a full moon to look back. That is a chosen habit, not an obligation established by astronomy. You can do the same on another recurring day. A calendar helps identify a moment; it does not establish that a phase determines success, mood or health. The source references explain the astronomical terms.
Source
Astrological interpretation for inspiration and self-reflection. Not a medical diagnosis or personal health advice.