In these studies, "happiness" is measured in a variety of ways. For example, the first study used these methods:

Satisfaction with life. The 5-item Satisfaction With Life Scale (SWLS; Diener, Emmons, Larsen, & Griffin, 1985) assesses the cognitive component of subjective well-being. Items (e.g., “In most ways my life is close to ideal”) are rated on a 7-point scale (1
strongly disagree; 7
strongly agree). In previous work, the SWLS had a 2-month test–retest correlation coefficient of r
.82, and coefficient alpha
.87 (Diener et al., 1985).

Campbell Well-Being Scale. The Campbell Well-Being Scale consists of nine semantic differential scales (e.g., boring–interesting, miserable–enjoyable, discouraged–hopeful) that provide an overall index of general well-being (Campbell, Converse, & Rodgers, 1976). The scale correlates with other measures of well-being and has acceptable internal consistency and reliability (Beckie & Hayduk, 1997).

Optimism. The widely used Life Orientation Test (LOT; Scheier, Carver, & Bridges, 1994) is an eight-item scale for assessing dispositional optimism. Scheier et al. (1994) reported an internal consistency reliability of

.82 and test–retest stabilities ranging from .56 to .79 across four time periods.

Depressive symptoms. Participants also completed the Center for Epidemiologic Studies Depression Scale (Radloff, 1977). Participants used a 4-point Likert-type scale (1
Rarely or none of the time [less than 1 day] and 4
most or all of the time [5–7 days]) to indicate how often during the last week they experienced each of 20 affective and somatic symptoms that characterize major depressive episodes. Total scores are the sums of scores from all 20 items. Radloff (1977) reported test–retest stabilities ranging from r
.67 (4 weeks) to r
.32 (12 months). Internal consistency was estimated as

.90 (Radloff, 1977).

Positive affectivity (PA) and negative affectivity (NA). The Positive and Negative Affect Schedule (PANAS; Watson, Clark, & Tellegen, 1988) measures general tendencies to experience positive (e.g., “proud”) and negative (e.g., “guilty”) affect. Participants used a 5-point Likert-type scale (1
very slightly or not at all; 5
extremely) to indicate how well each of 20 adjectives described “how they generally feel.” Coefficient alphas of the positive and negative scales range in the mid- to upper .80s (Watson et al., 1988).

Loosely, "pleasant emotions". Most dictionary definitions will give you something like this. There's probably a more specific meaning in psychology.