ATTQOTD: I generally only carry water (even if it's really hot/humid) if I'm going to be out over 60 min. If I'm not carrying liquids, I have a nice little waist belt to hold my phone and keys.
For liquids, I use a hydration vest (Salomon for me). The reservoir holds 1.5 L and I can fit two soft 0.5-L flasks in the front vest pockets, so I can hold various amounts. If it's an actual running hydration vest (not a hiking backpack for example), they are pretty minimalist and light. It's nice that they center the water on your back (hardly notice it) and you can drink from the straw without holding something in your hands.
The vest has sufficient stretchy pockets for my phone, some nutrition, a folded-up hat, etc., as needed, all within easy reach.
have never tried a hydration vest as I overheat easily,
Pro tip: depending on the outside temperature, freeze your water in the reservoir ahead of time or fill with lots of ice + your liquid. Then you're carrying a cooling system right on your back via your vest.
If you freeze the liquid ahead of time (completely solid), it can take quite awhile for it to thaw - longer than you might think - so it's a bit of an experiment for which outside temperatures you need to thaw it in time to drink it.
I will admit that filling the reservoir with ice cubes can be a little noisy until the ice cubes melt. This may be different for different styles of reservoirs.
A word about Salomon vests: the women's sizes tend to run small. Either get the unisex version or size up one in the women's size compared to what they suggest. (The arm holes run small - apparently they think all women have stick-figure arms.)