Yes and no (at least in US English, since UK English still uses "pudding" to refer to quite a few other dishes). "Custard" refers to any instance of thickening sweetened milk with eggs, which means it includes creme anglaise, many ice cream bases (which is really just creme anglaise, so this is kind of redundant), and stuff that has other thickening agents in there like starch (creme patissiere) or gelatin. "Pudding" refers specifically to thicker custards, as well as to some variants that don't have egg at all, such as panna cotta (milk+gelatin) or flour-based puddings. There's a lot of overlap, such that many people use them interchangeably, but they are still different terms.
Robbie Burns Day is on Tuesday. If there were ever a good time to try it, it's then. Finding good stuff can be a challenge, though. There aren't many restaurants that do it, and canned/frozen stuff can be kinda hit-or-miss (and you probably don't want to make your own from scratch).
I like them all anyway. Though 99% of the time I'll go for vanilla or another flavor over chocolate pudding. Panna cotta is probably my favorite dessert.