When we are out at restaurants we will sometimes reach for food on our plates and eat it but will not eat the same food if we put it in front of him.
I think this could be key.
I have a very good eater and he has fed himself since 6 months old BUT he has phases where he appears to not trust the food I give him. This is not something I have read about but my own theory based on what I have experienced with DS. Every time he has a developmental leap, learns something new, he questions the world around him again to see what has changed and what has stayed the same. Food is one of the things he 'learns' to mistrust every now and then with his increased awareness of the world (this is a good skill to learn, it could help him make a choice not to eat poisonous berries. Annoying behaviour often has a positive side to it too).
Long and short of it is mummy and daddy are like the court jesters, you are there to 'test the food for poison' before LO will trust it.
If he wants food off your plate then for a short while I would dish his up onto your plate and offer one or two pieces at a time by moving them onto his plate/tray. If necessary take it back off his plate and make a point of eating a bite of it, say 'yum that's nice' (you don't need to go overboard on this) and put it back on his plate. If he eats it don't make a big deal of it. If he doesn't eat it don't make a big deal of it.
I would stop trying to spoon feed, at 13 months he can pick up food and put it to his mouth himself and most likely wants to. I would also make a big effort to have family meals together and never feed him separately. Eating for humans is a sociable activity. We always eat more when we are with others. It also helps LO to learn about table manners and how things are done by watching you. I would also turn the cartoons and TV off. Again eating is a sociable occasion and some family togetherness and conversation at the dinner table is a valuable thing. Food can should be enjoyed, and shouldn't need distraction of cartoons.
As for your food going cold, well, I often have cold food too. DP and I try to keep our food warm whilst LO's cools down to enable us all to sit down to eat together, but often it isn't so warm by the time we are all at the table. But if yours is going cold because you are fussing over LO, then stop fussing
Stop trying to feed him and let him do it. If finger food is pretty new to him he will need to work through a phase of exploration to see what the texture, weight and feel of it is, plus how well it drops to the floor and if it's any good as hair gel. When he is hungry it will find it's way into his mouth. There will naturally be things he likes and things he doesn't. Mine hardly eats any meat or fish but will eat plenty of veg and carbs for example.
I know it's a lot more effort to have home made food and all sit down together but I really do think that is the way forward and it will be so worth it in the future when you have already set up the expectations of family time at meal time.
Good luck with working through this phase. It must be difficult for you when you have such long hours to work too xxx