The crossbar of a t is not a serif, but those of the capital I are definitely serifs.
Only on computer screens it is possible to confuse serifs with crossbars, because of the very low resolution, which forces the increase of the width of a serif to 1 pixel, possibly making it as wide as a crossbar.
To convince yourself that capital I has serifs and not crossbars, just look at high-resolution photos of some Roman imperial inscriptions, like that on Trajan's column, which are the gold standard for the design of the capital letters in serif fonts.
Most letters of the Latin script are made of 3 elements, thick lines, thin lines and serifs. The width ratio between the thick lines and the thin lines is called the contrast of the font.
Serif fonts normally have a higher contrast and sans serif fonts not only have no serifs, but they also have no contrast or only a low contrast.
Serifs are even thinner than the thin lines (which include some of the crossbars), except in sans serif fonts (which have no serifs) and slab serifs fonts (where the serifs are as thick as the thin lines).
Both the sans serif and the slab serif fonts are fonts typical for the 19th century after the Napoleonian wars, when they were used mainly for advertising, where they attracted attention due to their anomalous serifs and they also allowed a lower cost by using cheap paper and printing machines, which would not have rendered well the standard serif fonts.
In several programmer fonts, where most characters are sans serif, a few characters are made slab serif, i.e. with serifs that are as thick as a crossbar, with the purpose of distinguishing them clearly from similar characters. Thus capital I is made with thick serifs looking like crossbars, even if that is not the standard capital I shape. The reason is less to distinguish it from l, which should have a low hook even in sans-serif typefaces, but to distinguish it better from vertical bar, which is important in programming languages.
Moreover, because such programmer fonts are fixed-pitch, a few narrow characters have slab serifs that do not exist in variable-pitch fonts, in order to avoid excessive areas of white space between letters. Such slab serifs added for blackening are put at the top of the small i, j and l letters, not only on capital I (but on the small letters the slab serifs are unilateral, not bilateral, like on capital I). Such extra slab serifs on the narrow characters are inherited from the type-writing machines, where they had the purpose to diminish the pressure of the hammer hitting the paper, to avoid making holes in the paper.