This is a continual question. Austin-Healeys, and other European cars, did not get redesigned every year for the "new model year" as did American cars. Instead, they just made running changes, implementing them whenever they were ready. Likewise, new models were introduced when ready, and were not kept secret for an unveiling in September every year, as with American cars.
Therefore, the year that your Healey was registered/titled is not very important. Likewise, the year that it was made is not very important. The model/series is what is important. For example, from the year of your Healey, 1964, I don't know if it is a late BJ7 or an early BJ8 (those being the two likely possibilities).
As an aside, have you ever noticed that a disproportionate number of BJ8 series cars (that's the 3000 Mark III for the uninitiated) are registered/titled in 1967? If you check the date of manufacture, you'll find that a whole lot of these cars were actually made in 1966, and occasionally you even find one made in 1965 and titled as a 1967. The reason for this is that basically any car titled from September 1966 onwards was assumed, by American DMV offices, to be a "1967 model." After all, the new American cars being sold beginning in September were "next year's models" and were titled as such. This practice ended, for the most part, for Austin-Healeys in 1967. Since they did not meet the U.S. emissions or safety standards that came into effect on January 1, 1968, they had to be registered/titled as 1967 models, even if they were purchased at the local BMC dealer in 1968! If a 3000 Mark III was sold n September 1968 after sitting on the dealer's lot for, say, a year or more, it might have been titled as a 1969 under different circumstances, but since it didn't meet the standards for 1968 (or later) models, it HAD to be registered/titled as a 1967 to be legal. This is why it seems that 90 percent of BJ8 series cars, which were built from late 1963 through the end of 1967, are registered as 1967 models.