Perhaps the most critical question to consider is where your fireplace will go. For small patios, you'll need to measure the space to make sure the fireplace you want will fit without crowding out other furniture. You should also avoid placing the fireplace too close to any steps or ledges, especially if there isn't a railing.
If the largest location on the patio or in a patio wall is far from your tables and chairs, though, it might be worth it to choose a smaller spot. A faux fireplace that is far from your guests or out of their line of sight might not have the desired effect on your patio's atmosphere.
You can also try a corner fireplace if there's not enough room elsewhere on your patio. These are designed to fit seamlessly in the corner of a room and also work well for outdoor spaces as long as they are rated for outdoor use.
Standalone or Built-In
Most outdoor fireplaces are designed to be independent of any walls, other than needing an extension cord for electricity. This makes them easy to move around as necessary, including bringing them indoors if needed during the winter months.
However, you might find that it's worth improving your home's value by installing a fireplace into a wall. Building a fireplace into the same wall as the door is usually inconvenient, but building a separate low wall on one side adds dimension to your patio.
The best option for you will partly depend on how much space you have. Standalone fireplaces often require a certain amount of free space behind them as well as in front of them, so they shouldn't be placed up against railings or walls.