Does play a personality mean play acting, or does playing a personality just mean choosing actions based on the personality - in addition to the chosen class? I.e., what would a fighter with these personality traits do in this situation?
The two are related because - at least in some RPGs, like classic D&D - "personality traits" aren't something that figures in the game except by play acting of some sort. Eg AD&D just doesn't have any way to express "hates kobolds" as part of the function/capabilities/responsiblities of the character. (Contrast 5e, which does, via the personality/inspiration rules.)
Thinking more about the personality thing in the classic game - there is
not the least suggesion - not at all, not one bit - in Gygax's PHB that I might make choices for my character based on personality, as opposed to based on what is expedient from the point of view of dungeoneering.
Here are some of the key passages (PHB pp 107, 109):
Now assume that a game is scheduled tomorrow, and you are going to get ready for it well in advance so as to have as much actual playing time as possible - no sense in spending precious adventuring minutes with the mundane preparations common to the game.
First get in touch with all those who will be included in the adventure, or if all are not available, at least talk to the better players so that you will be able to set an objective for the adventure. . . . some firm obiective should be established and then adhered to as strongly as possible. Note, however, that inflexibility or foolish stubbornness is often fatal. More about that a bit later.
Once the obiective has been established, consider how well the party playing will suit the needs which it has engendered. Will the characters have the means of accomplishing the goal? Is it well-balanced, so that it can cope with typical problems expected in the fullfillment of the objective? Will it be necessary to find mercenary non-player characters or hire men-at-arms in order to give the party the necessary muscle? Is any special equipment needed? When agreement regarding these and any similar questions has been reached, each participant must ready his or her character, but preparations must be made with the welfare of the whole group in mind.
Co-operation amongst party members is a major key to success, particularly when the characters are relatively low-level. . . .Co-operation must begin when the party prepares for the adventure and continue through safe return to base and division of spoils - including the special treatment required for any unfortunate characters cursed, diseased, maimed, or killed.
Each character has a selection of equipment which he or she will carry on the adventure. Particulars should be given to the party if any equipment is possibly redundant, newly conceived, or of possible special use considering the established goal for the adventure. In like manner, spells must be selected in co-operation with other spell-users in general, so that attack, defence, and assistance modes will be balanced properly and compliment the strengths and weaknesses of the party as a whole. Characters must know each other's strengths and weaknesses, physical and mental, in order to meet the problem posed with the correct character or combination thereof. Does the group have sufficient equipment of the elementary sort to meet both expected and unexpected challenges (ropes, spikes, poles, torches, oil, etc.)? Are we burdening ourselves with too much because of simple duplication (too many torches, everybody has a 10' pole, and so on)?. Do we have as broad a spectrum of spells as possible so as to be able to have a good chance against the unexpected, considering the objective and what it requires in spells? Is there some magic item which one of the party members possesses that will be of special help, or general assurance of survival, in this adventure? All this should be done before play begins, for it is time consuming, and the readying of a party can require several hours if there are more than six characters involved. . . .
Your party has an objective, and wondering monsters are something which stand between them and it. The easiest way to overcome such difficulties is to avoid the interposing or trailing creature if at all possible.. . .Do not be sidetrocked. A good referee will have many ways to distract an expedition, many things to draw attention, but ignore them if at all possible. The mappers must note a11 such things, and another expedition might be in order another day to investigate or destroy something or some monster, but always stay with what was planned if at all possible, and wait for another day to handle the other matters. This is not to say that something hanging like a ripe fruit ready to be plucked must be bypassed, but be relatively certain that what appears to be the case actually is. Likewise, there are times when objectives must be abandoned. . . .
All members of the expedition should be ready and willing to part with any goods, money, and magic items in order to save lives. Failing that, each should be willing to fight to the death to assure the survival and success of the party. This will happen when mutual trust exists. . . .
Generally evil characters, particularly chaotic evil ones, are prone to be troublesome and hurtful to the party. They should accordingly be shunned when possible. Selfish neutrals are similar to evil characters, but their price is usually easier to meet, and it is therefore easier to integrate them into an expedition which will depend on co-operation for success. The character of good alignment who is basically unco-operative - often acting as an evil or (selfish) neutral would - is another matter, for such players usually join under the pretense of being helpful and willing to act in the best interest of the party. Undoubtedly the best way to take care of such players is to expel them from the group as soon as circumstances permit.
Nothing is said anywhere about character personality. The idea that I might not pack rope because I'm playing a forgetful character; that I might choose only fire spells because my wizard is a pyromaniac; that my cleric might refuse to heal some PCs because she doesn't like their behaviour; etc - these are all completely absent.
Which is to say, the "role" is being conceived of not as a character's personality, but as a function or capacity that the character brings to the game.