What is the difference between Type 1 and Type 2 errors and the severity of the error?

Type I Error:

A Type I error is often referred to as a “false positive" and is the incorrect rejection of the true null hypothesis in favor of the alternative.

In the example above, the null hypothesis refers to the natural state of things or the absence of the tested effect or phenomenon, i.e., stating that the patient is HIV-negative.
The alternative hypothesis states that the patient is HIV positive.
Many medical tests will have the disease they are testing for as the alternative hypothesis and the lack of that disease as the null hypothesis.
A Type I error would thus occur when the patient doesn’t have the virus, but the test shows that they do. In other words, the test incorrectly rejects the true null hypothesis that the patient is HIV-negative.

A Type II error is the inverse of a Type I error and is the false acceptance of a null hypothesis that is not true, i.e., a false negative.

A Type II error would entail the test telling the patient they are free of HIV when they are not.
Considering this HIV example, which error type do you think is more acceptable? In other words, would you rather have a test more prone to Type I or Types II errors? With HIV, the momentary stress of a false positive is likely better than feeling relieved at a false negative and then failing to take steps to treat the disease.
Pregnancy tests, blood tests, and any diagnostic tool that has severe consequences for a patient's health are usually overly sensitive for this reason – they should err on the side of a false positive.
But in most fields of science, Type II errors are seen as less severe than Type I errors. With the Type II error, a chance to reject the null hypothesis was lost, and no conclusion was inferred from a non-rejected null.
But the Type I error is more severe because you have wrongly rejected the null hypothesis and ultimately made a claim that is not true. In science, finding a phenomenon where there is none is more egregious than failing to find a phenomenon where there is.