An uptrend-line, by definition, needs to touch a minimum of 2 low points but the more low points it touches the higher the confirmation of the trend. Note that many times you might get the low of the day crossing below the trend-line but it then closes back above the trend-line - this would not be considered a break of the trend-line as it needs to close below the trend-line.
A downtrend-line, by definition, needs to touch a minimum of 2 high points but the more high points it touches the higher the confirmation of the trend.
The trend-line is used to give a visual indication of the trend. If you have a trend-line drawn that touches at least 2 points and then try to redraw it when there is a close below it then you are curve fitting the trend-line to suit your biases, and that will not end up well if you are using it to trade with. Also note that sometimes you might have an existing uptrend-line drawn, and then the price starts moving higher at a faster rate - so in this case you might consider drawing a secondary steeper uptrend-line. If the steeper uptrend-line is broken you are still in the original uptrend until that line is broken. The same would be the case with downtrend-lines.
You have picked up on one problem with trend-lines, different people can place different trend-lines on the same chart based on their biases. For example, some can use the low of the day for the points the trend-line touches whilst others may use the closing prices. You should choose what you want to use as your criteria and then be consistent each time you draw a trend-line.
For this reason, a better way to determine the status of any trend is to use price action itself. The definition of an uptrend is higher highs and higher lows. The definition of a downtrend is lower lows and lower highs. In using this definition to determine the trend there is no bias as you are using price action itself, and not any indicator derived from the price action and in which you can change parameters to get different results.
So in using this definition you can determine the start and end of an uptrend and downtrend. As a downtrend is defined by lower lows and lower highs, when you get a higher low and then confirmed with a higher high, the downtrend would have ended and an uptrend would have started.
Similarly, as an uptrend is defined by higher highs and higher lows, when you get a lower high and then confirmed with a lower low, the uptrend would have ended and a downtrend would have started.
Many times a trend may end without the opposite trend commencing and the price simply moves sideways in a trading range for a period before the original trend continues or the opposite trend commences. Also, you can have a trend within a trend. The price might be in an uptrend on the daily chart, however, on the weekly chart it is in a downtrend. So the trend will depend on the time frame you trade in and period of time you hold on to positions. It is no use looking at the weekly trend if you hold positions for a few days, and it is no use looking at the hourly trend if you plan to hold positions for months or years.
The more charts you look at to determine the status of any trend and draw in trend-lines, the better you will get at reading trends and changes in trends. Good luck.