Here's my understanding of computing the signs in Morse homology (following the book by Audin and Damian). Let $f$ be a Morse function on a manifold $M$ with a negative pseudo-gradient $X$. Let $W^s(p), W^u(p)$ denote the stable and unstable manifolds of a critical point $p$ and we orient all the $W^s(p)$ (which being diffeomorphic to a disc is orientable and contractible).
For critical points $a, b$ set $\mathcal{M}(a, b) = W^u(a)\cap W^s(b)$ and let its quotient by $\mathbb{R}$ be denoted by $\mathcal{L}(a, b)$.
For $p\in\mathcal{M}(a, b)$ we look at $$0\to T_p\mathcal{M}(a, b)\to T_p W^s(b)\to N_pW^u(a)\to 0$$ The normal bundle $NW^u(a)$ is trivial because $W^u(a)$ is contractible, so $$N_pW^u(a) = N_aW^u(a) = T_aM/T_aW^u(a) = T_aW^s(a)$$ is oriented. So, a basis $B$ of $T_p\mathcal{M}(a, b)$ is positively oriented if orientation of $N_pW^u(a)+B$ gives the orientation of $T_pW^s(b)$.
For the quotient, I use $X$ and say that a basis $B'$ of $T_p\mathcal{L}(a, b)$ is positively oriented if $X+B'$ gives the orientation on $T_p\mathcal{M}(a, b)$.
So, if I put it all together, I should say that $B'$ is positively oriented if $N_pW^u(a)+X+B'$ gives an orientation of $T_pW^s(b)$.
Now I look at $\mathbb{R}P^2$ and the function $f = ax^2+by^2+cz^2, a<b<c$ and let $u$ denote the maximum, $v$ the saddle point and $w$ the minimum. $NW^u(u)$ is the $0$ bundle, so an orientation is just a sign ($+$). So, what I get is that a flow line from $u$ to $v$ is oriented $+$ iff it agrees with $X$, but this gives the two flow lines opposite signs (because $W^s(v)$ is an interval going from $u$ to $u$ through $v$).
I don't know where I'm going wrong and any help is greatly appreciated.