Put two small plates in the freezer. Place the chopped apples in a saucepan with 60 ml water and the lemon juice. Simmer gently for a few minutes, stirring occasionally, until beginning to soften but still holding their shape. Set aside with their cooking liquid.
Combine the white granulated sugar and remaining 40 ml water in a separate, deep, heavy-based saucepan. Heat gently, stirring until the sugar dissolves.
Once the syrup boils, remove the spoon. Do not stir with a spoon or spatula while the caramel is boiling. Cook uncovered over a medium heat until amber, gently swirling the pan if it colours unevenly. Remove from the heat.
Carefully add the warm apples and their cooking liquid in small additions, allowing space for bubbling and steam. Return to a low heat and stir gently until any hardened caramel dissolves.
Add the light brown sugar and optional cinnamon. Stir over a low heat until the sugar has dissolved.
Increase the heat to maintain a steady bubble. Cook uncovered, stirring regularly and more frequently as the jam thickens. Begin testing after about 10 minutes; the apples should be tender and the surrounding syrup thickened.
Remove from the heat and spoon a little jam onto a cold plate. Cool for one minute, then push with a finger: the syrup should move slowly and wrinkle slightly, without watery liquid spreading around it. If needed, cook for another 2–3 minutes and test again.
Remove from the heat and stir in the butter, salt and vanilla. Rest for 5 minutes, stir to distribute the apple pieces, then spoon into clean, warm, heatproof jars. Fit the lids, cool and refrigerate promptly.