First, as a minor note, you're not seeing arteries, you're seeing veins. Hence getting more "vascular" due to shed body fat.
I can almost guarantee that working out your core is not the reason for gaining 10lbs. Maybe, and that's a MASSIVE maybe, side planks led to your obliques being developed and added maybe a lbs of solid muscle which could lead to a blockier waist (not fatter, but obliques are on the sides of your torso) but that's a very long shot.
What's way more likely is that your back workouts are simultaneously working your lower back and since you're not used to weight training, the majority of the muscle you added is around this region. What do you do for back? Deadlifts, bent over rows, and lots of other back exercises could do this. If not, it is almost certainly due to your diet. Can you update your post to give us your diet? If you've followed the exact same diet for a very long period of time, your body may have adapted and since you weigh less, if you didn't adjust your calories accordingly, you may be eating an excess..or you might have followed a crash diet, or you changed your diet to manipulate carbs, glycogen, and therefore cortisol and this may be causing fluctuations in the weight...very hard to tell unless I know your diet and your daily habits as well as workout routine.
As for your actual question, while this is more or less different for everyone due to genetics, there are some obvious tells. Your veins are certainly one of them, and this isn't limited to your forearms (I actually got a few bicep and shoulder veins sometime before clear forearm veins). Given that it is different for everyone, the safest way would be to just compare some past pictures of yourself to your present self. Your waist size isn't really the best indicator after a while. The most obvious way is to just look at yourself in the mirror (sideways) and see how much your belly sticks out compared to before, how much love handles you have...etc. From my experience the muscle that sticks out the most as you lose fat tended to be the biceps (there is a major vein on your bicep that'll become more visible at even moderate-high bf levels) and the little "cut" or seperation between your shoulders and your arms.
Everyone has a unique body, you should try not to compare yourself to anyone but yourself. Most common/obvious places for the purposes of your question though, are abs, arms including forearms, your head (this is actually a pretty big one, and your close ones will have noticed), and in some cases, your hands and feet.