I’m concerned that with my meditation, stretches AND short exercise session/run in the morning, I’m extending the length of my morning routine too much because it takes me some time to cool down after my run before I can shower and eat. Is it better to meditate after exercise? That way I can start my exercises immediately after waking up and having my water.

I would definitely use meditation as part of your cool down process. If it’s going to be apart of your routine it needs to be an advantage not a hindrance. Meditation is way of reviving but also relaxing your body so doing it before an exercise can be draining but doing it after will help slow your heart rate and ease you into your shower and breakfast.

For me, what works better is to keep the morning routine as simple as possible to have more control of the time.
About meditation, in my opinion (I'm not an expert about this topic), is better do it in the night because two things: the first one is that meditation can calm down your body and give you a better sleep; the second one that when you wake up you body just have finished the rest, so you will not get all the benefits of meditation in this moment.

For me, do the exercise after the water drink helps me to not procastinate on my daily routine

I think you have a distorced meaning of "meditation". Meditation is not only "staying still" or "sit down in the lotus position". Mediation is pure awareness of what you are doing in the present moment. You can also meditate by moving and stretching. So, you can meditate also during your exercises.

Hi,
It is best to do your exercise in the morning as soon as you wake up. Meditation can be done anything during the day, but it's more effective after a physical workout. Make sure to drink a glass of water as soon as your awake to hydrate your body and prepare your muscles for the short run.
(Water, exercise, water, meditation, shower, eat and go on to your daily activities.)

Personally,I feel it's better to exercise and then meditate. Meditating after an exercise can help us to also cool down and will save some time.

I hope this helps. I totally understand the thought you having a "big" morning routine, as that puts me off because I just think it will take to much time to get to the start of to-do list. I would advise you to move your meditation to another time of the day, or even before bed time, so you can reflect each night about the day and what you hope for the next day. Also, if you prefer to drink water and exercise, you can take a few seconds while you drink your water to be mindful of the amazing day, while you deep breath. Is not a wrong or right thing, is just the way if helps you get through the day better. Hope it helped ✌🏻