I think it may not be either issues. I think the data is likely correct.
Subtropical, Extratropical, Lows and Disturbances can all have high wind intensity, but that doesn't mean they are hurricanes. A storm needs to be determined to be a tropical cyclone before it can rise to the level of a hurricane (based on wind speed/intensity). Definitions of types of storms here: https://www.nhc.noaa.gov/aboutgloss.shtml
https://www.nhc.noaa.gov/data/hurdat/hurdat2-format-atl-1851-2021.pdf
HU (Spaces 20-21, before 4th comma) – Status of system. Options are:
TD – Tropical cyclone of tropical depression intensity (< 34 knots)
TS – Tropical cyclone of tropical storm intensity (34-63 knots)
HU – Tropical cyclone of hurricane intensity (> 64 knots)
EX – Extratropical cyclone (of any intensity)
SD – Subtropical cyclone of subtropical depression intensity (< 34 knots)
SS – Subtropical cyclone of subtropical storm intensity (> 34 knots)
LO – A low that is neither a tropical cyclone, a subtropical cyclone, nor an extratropical cyclone (of any intensity) WV – Tropical Wave (of any intensity)
DB – Disturbance (of any intensity)
So if a storm does not meet the requirements to be classified as a tropical cyclone, regardless of wind speed, it will never have a status of hurricane.
For example, you've probably experienced wind conditions 34 - 47 knots, which is a gale, unless the wind was associated with a storm that was already determined to be a tropical cyclone (based on additional criteria aside from wind).
Or you may have been in a winter Nor'easter (extratropical storm that can have winds over 65 knots but isn't a hurricane). I don't know the measurements behind it, but reading the definition and the link below it appears there are multiple characteristics/metrics used to determine if a storm is a tropical cyclone.
Category > 0 had me stumped also at first, but then I realized the categories are based on a wind scale. Non-tropical cyclone storms also get assigned categories based on wind in the data. At first glance they appear to coincide with the Saffir-Simpson Wind scale, but I don't know that for sure. Having followed weather reports closely as a sailor, though, I've never heard NOAA refer to a category 1 gale or category 1 nor'easter, so I think NOAA only uses categories to describe hurricanes. https://www.nhc.noaa.gov/aboutsshws.php (see my second comment below - just realized the category data did not come from the original file).
luis.df <- storms %>%
filter(name == "Luis") %>%
select(year, name, category, status, wind)
47 | 1995 | Luis | 2 | hurricane | 95
48 | 1995 | Luis | 2 | hurricane | 90
53 | 1995 | Luis | 2 | EX | 85
54 | 1995 | Luis | 2 | EX | 95
55 | 1995 | Luis | 3 | EX | 105
56 | 1995 | Luis | 2 | EX | 90
57 | 1995 | Luis | 1 | EX | 75
58 | 1995 | Luis | 0 | EX | 60
Hurricane Luis, for example, was an extratropical storm at some point, with wind categories 1, 2, and 3, but the other characteristic of the storm at that time did not meet the criteria for a tropical cyclone anymore, even though the winds were often higher than when it was a tropical cyclone.
So I guess the take-away would be not to filter the data based on category thinking you are only going to get hurricanes.