ている
te-iru
ている is not 'is doing'. It is a switch left on.
The picture A switch flipped on over the verb. Either it is still running, or it was flipped once and has stayed on. ている says the switch is on; てある says someone deliberately left it on.
Where the picture shows up in ている
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still running
今、昼ご飯を食べています。
ima, hirugohan o tabete imasu
I'm having lunch right now.
The chopsticks are still going up and down over the bowl, and the switch above them is lit.
-
left on for good
母は大阪に住んでいます。
haha wa ōsaka ni sunde imasu
My mother lives in Osaka.
The light over the Osaka house has been on for years and nobody has turned it off.
Plain 住みます sounds wrong. Living somewhere is a switch that stays on, so 住んでいます is the normal form.
-
the result is still on
電気がついています。
denki ga tsuite imasu
The light is on.
You can't see who flipped it or when — only the bulb, burning now.
-
a habit kept switched on
毎朝、六時に起きています。
maiasa, rokuji ni okite imasu
I get up at six every morning.
The same switch clicks at six every morning, leaving a neat row of marks down the calendar.
-
it became so and stayed
姉は結婚しています。
ane wa kekkon shite imasu
My sister is married.
A switch flipped once on the wedding day and still glowing today.
結婚しました is the wedding as an event; 結婚しています is being married now.
All of it in one picture
ている is not a tense; it is the question of whether the switch is lit. If the chopsticks are moving, the action is running; if the light over the Osaka house has stayed on for years, that is where she lives; if you cannot see who flipped it and only the bulb is burning, it is the result left behind. A row of clicks at the same hour is a habit; one click on the wedding day, still glowing, is a present state. All five are the one picture: the light is on now.