First things first, until you get this corrected stop using your 1500W heater! You're seriously overloading your 16 AWG extension cord. That cord is only good for about 870W (source). Very likely the only reason it hasn't melted is that you're using it outside in the cold, so it can shed its excessive heat more easily than normal, but it's only a matter of time before it warms up outside faster than you remember to turn the heater off, and suddenly your grass is on fire.
Now that you've posted pics of your main panel, there's another issue I want to bring to your attention: you main panel is currently configured in a way that violates both its instructions and the electrical code. Unfortunately, this is not a situation of old work that's grandfathered today, as installing it in that manner was never legal. That means that very likely unpermitted work was done on this panel at some point, and therefore an inspector that sees it has the authority to make you rip it out. This could happen even if they're only coming over to check out your new extension of your existing 15A circuit.
The good news is that the panel itself is perfectly good, despite its age. It's a Murray, which is now owned and supported by Siemens, and breakers for it are widely available, including AFCI, GFCI, DFCI, and other exotic types. There's no need to replace the panel itself.
The trouble comes from how the circuit breakers are arranged. If you look at the diagram in the instructions, you can see that tandem breakers are only allowed in the bottom section, and not in the two upper right positions in the bottom section. Code requires that you follow this labeling, because that's the only configuration the manufacturer & UL safety tested it in. So what you need to do to bring this panel into compliance is to rearrange the breakers such that all of the tandem breakers are in the slots they're allowed to be in.
Unfortunately, doing this won't be as easy as it is on many panels, because this is an old "rule of six" panel that you can't fully deenergize yourself -- you'll need your utility company to pull the meter to kill power to this panel. This is an obsolete design, where the entire top section of the panel is always on, and to fully disconnect your house, you have to flip off every breaker in that top section -- there's no main switch to deenergize everything. The "rule of six" is from older code versions, where you're not allowed to have more than six switches to disconnect all power from a building.
In 2020 code for new installations, it's actually now required to have a single outdoor disconnect. This is so that if your house is on fire, firefighters can quickly turn off all power, so that they can chop through your walls with their big axes to rescue people as needed without fear of getting electrocuted by live wiring inside the wall. When you have your main panel disconnected for breaker rearrangement, strongly consider adding an outdoor disconnect: you'll be compliant with the latest codes, have improved fire safety, and you'll be able to fully deenergize your panel whenever you want. It also opens up Option 3 below.
Now, on to your main question...
Unfortunately, heating with electricity takes a lot of power. That's why electric ovens and water heater and such always have dedicated circuits with thick cables running to them. A 1500W heater, like you have, use up the entire capacity of a 15A circuit all by themselves. (*1)
Since you're wanting to heat this space with 2000W, running a single 15A circuit like you propose is insufficient for your heating needs, much less your lights and computers and whatever else is connected to this circuit in your house. Why didn't your breaker trip? When it's only a little overloaded, it takes a while -- the wires in your walls are slowly overheating, and the thermal trip mechanism in the circuit breaker is heating a little faster, such that it'll trip before the wires get too hot. But if it's working properly, it will trip eventually. If it's not, then you could start a fire in your walls.
However, one 15A circuit is plenty to run your lights and computer if you find a different way to heat the space. What I'd recommend is getting a propane or kerosene heater, and using that instead. Or if your home has easily accessible natural gas piping, you can run a gas pipe through the same trench you're using for electrical, and use that instead. This will have the advantage of being cheaper and more efficient to run, too: electric resistive heat is the most expensive kind, since you're basically paying for a power plant to burn some fuel, vent half the heat through their smoke stacks, and send the other half to you as electricity. If you burn the fuel yourself, you can get nearly all of the heat.
Just like your electric heaters, a fuel based heater must be rated for indoor use, and located sufficiently far from anything flammable. Additionally, they'll need a source of combustion air and a place to exhaust the fumes. If your shed is reasonably well ventilated, you can probably get away with the kind that uses the air from the room for combustion (often sold as "ventless"), but some of the bigger and/or the more efficient ones will want you to route both an intake and exhaust pipe outside -- if you want this kind, plan for this before you cover the walls. And in all cases, follow the instructions of the heater to the letter -- that's how they've been tested to run safely. And it would be a good idea to also add a carbon monoxide detector to your shed, to warn you in case anything goes wrong.
If you go this route, for the electrical circuit extension, you can follow Ed Beal's good advice for how to actually extend the circuit. IMO this is the best solution in terms of cost and efficiency.
Since your main panel is out of spaces, the easiest way to gain more, and therefore gain slots to add a new feed to your shed, is to remove a couple of breakers (or one two-pole) in the main panel, and relocate them to the subpanel. Which one you choose depends in part on where you want the new subpanel -- for example, if you want it outdoors on the wall of your house, you might put it next to your outdoor AC unit, and relocate the AC breaker there. Then you can also run the shed circuit from that location, though the exact options will depend on your house's layout.
This option dovetails nicely with correcting the problems in your main panel. With this route, you'd install a panel outdoors, ideally near your electric meter, which will become your new main panel. This panel will contain 2 breakers; one to your current panel (which is now a subpanel), and one to your shed. This arrangement has the advantage that you can kill power to either building independently, from outdoors, which is very helpful for firefighter access. It also solves the problem of being able to deenergize your rule-of-six panel nicely, and brings your installation up to 2020 code. Note that this may require some rearrangement of the grounding within your main panel, and possibly some new ground rods and/or conductors. IMO this is the best long term solution for future expandability.
Whichever option you choose, I'd recommend you also add a second conduit to your shed for communications cabling as well, such as ethernet. You can't run most data cables down a power conduit, and adding a second conduit is cheap compared to the cost of digging the trench. Even if you're happy with your wifi performance today, future generations of wifi won't necessarily go through walls as well (that's the tradeoff for higher speeds) and your neighbors might set up access points which interfere with your signal. Adding the option for ethernet now is a very cheap way to future-proof.
(1. Technically, 120V*15A=1800W is the limit of a 15A circuit, but only for things that are intermittent. For something that may be on constantly, like a heater, 1500W is the limit.)
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