They also make BBQ lights that attach to the side of the bbq, they can be battery, or they do make ones that plug in.

https://www.amazon.com/dp/B01F7RO4YS?tag=duckduckgo-d-20&linkCode=osi&th=1&psc=1

also this https://www.amazon.com/Umbrella-Lighting-Cordless-Operated-Umbrellas/dp/B07Q27HCD6/ref=sr_1_1_sspa?dchild=1&keywords=umbrella+light&qid=1592526900&s=lawn-garden&sr=1-1-spons&psc=1&spLa=ZW5jcnlwdGVkUXVhbGlmaWVyPUExWEkzWDFYNUc5VUc0JmVuY3J5cHRlZElkPUEwOTUwMjM3MTVRRVZKS1I5R1VCTiZlbmNyeXB0ZWRBZElkPUEwMTc5NTM3MktYS1JSQTJMRTFTViZ3aWRnZXROYW1lPXNwX2F0ZiZhY3Rpb249Y2xpY2tSZWRpcmVjdCZkb05vdExvZ0NsaWNrPXRydWU=

Now if you want to spend $$ landscape lighting would really help.

you could change the patio door lights to more of a warehouse-style so they shine down this way the bbq would be better lit.

As solutions can range wildly in price, quality and appearance, with your question should be a rough budget figure that you are trying to keep within. At the low end would be patio string lights that are run generally around the patio perimeter or across overhead, from poles that are installed permanently, seasonally or for an event. The more peramanent it would be, the more expensive it would be. It is many small light sources that distribute the light more or less uniformly and looks festive. This "system" is often comprised of poles that are installed in planter tubs that are weighted with soil. There is not room on the patio for such tubs so you'd need to affix them somehow to the ground outside of the patio.

Another possibility would be a 10' or 12' yard pole down-light, probably located on the grill side, just off the patio edge, about equivalent with the landing of the steps. If more light was desired, a second pole could go on opposite side of patio. There are a variety of fixture styles. Would be higher quality, permanent and more expensive.

A third possibility would be to span the patio, over the landing area, with a 4-post pergola and conceal lighting within the upper "bridge" section. This would be the most expensive option as it requires construction and lighting installation. But it would allow for the light sources to be concealed and only the effect of lighting seen. There would be a spreading pool of light.

Probable cost range: a couple hundred $ to a few thousand.

It should be easy enough to install a light above the BBQ but the dining area is another issue. I do not believe post lights will give you enough lighting there . You could look into solar LED lamps for the table but IMO a better option would be a pergola built there with power run to it so you can install lighting. I do not sea a sitting area out there but I think it would be a nice addition to your outdoor space . BTW there are solar lights for umbrellas that might be the answer if a pergola is not in the plans.