With regards to the field of philosophy, are there any notable precise definitions of "pleasure" and "happiness"?
How does "pleasure" compare to "happiness" and vice-versa?
I wouldn't say there are "widely accepted (precise) definitions" for the two terms you mention. I would say there are several different well-known accounts that deal with relationship.
The first one I would recommend is from Aristotle's Nicomachean Ethics. For Aristotle, the end goal of life is eudaimonia -- a word that can and has been translated as "happiness." At the same time, this happiness also refers to a certain form of flourishing. Thus, Aristotle takes pain to explain that this is not identical to happiness nor is it necessarily identical to what agents take pleasure in. Instead pleasure/pain are signals that operate correctly in the virtuous person (evil pains them; good gives them pleasure), but that operate poorly in lesser mortals (where evil sometimes gives them pleasure; good sometimes gives them pain). The first part of this argument is BooK I. The concern with pleasure/pain happens in Book II. (I've written a little bit on this and continue to do so, but for more famous considerations, I would suggest consulting J.L. Ackrill's brief primer on Aristotle).
In terms of other accounts, we can also see differing treatments in the Epicurean philosophy and in the Stoics. For the Stoics, pain and pleasure signal nothing about morality except insofar as experiencing any negative feeling due to one's circumstances reflects a character that lacks ataraxia. Conversely, the Epicureans sought to optimize pleasure in their lives.
Moving towards more contemporary thought, the Stoic and Epicurean positions recur as the Kantian and Millean accounts. In John Stuart Mill's account, pleasure and happiness occur as synonyms. For Kant, happiness (in its rational form) is good, but otherwise the pursuit of pleasure is a sign of moral failing (see for instance his treatment of sex in the Metaphysics of Morals).
There's a decent amount of contemporary literature on the subject -- centered on what is required for people to be happy. But "happiness" is generally called "well-being" instead in the contemporary literature to avoid self-helpers molesting the research. I believe a large part of the impetus is Derek Parfit.
I can recommend at least two authors in this arena. Chris Rice who writes about objective list theories and Christopher Gowans who is also working on well-being. Objective list theories propose that people are happy (have well-being) when they have some or all of the goods on an objective list. This stands in contrast to subjective theorists who maintain that happiness is had in having what one wants.
I don't know if that answers your question, but that's the literature as far as I know it.
Firstly, the two English words can be understood differently through times and geographies. For example, eudaimonia means literally well-classed spirit and is translated as "happiness". The difference is too remarkable. "Pleasure" in hedonism can be used interchangeably with "happiness".
So it is safe as long as concerned in philosophy, no clear lines are important. It depends on your philosophical position to go further into concepts such as eudaimonia, provided they are not precisely "happiness". You can't understand a theist being atheist.