Painting with a very broad brush, Quality is a measure of the difference between all the background noise and the signal level itself. Strength is simply a measure of how large the signal is, whether that's clean or noisy.
Let's take your 45dB minimum strength as a starting point. It's a guideline minimum to get all the stations, strong or weak. It is possible to get perfectly decent reception at less than 45dB, but it doesn't leave much wiggle room if say the atmospheric conditions change or something else temporary causes the signal level to drop.
A signal level of 55dB at the set's aerial socket is a sensible target point. It gives a reasonable amount safety margin for fluctuations, and it isn't going to oversaturate a tuner.
65dB is the recommended upper limit, but there's still another 5dB or so headroom for most TVs.
All the above relies on there being a big enough difference between the background noise and the carrier signal. It's perfectly possible to have 65dB of strength, but a lousy carrier to noise ratio (CNR) which means pixelated reception. This sort of thing happens when there's a problem say with the aerial alignment resulting in a poor CNR, and then the owner adds an amplifier at the back of the TV to 'boost' the signal. Electronic amplification adds its own noise, and so makes the CNR worse, while making the total signal (including the old background noise) louder.
Passive splitters reduce the signal Strength but add no noise.
Digital signal levels is really quite a big topic. We measure CNR and the Bit Error Rate (BER), but there are different guidlines for some of this data depending on the signal modulation such as 64 QAM and 256 QAM. It's easy to get lost down the rabbit hole.
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