Yes, "an LFO applied to pitch" is the same basic structure as "FM synthesis". In both cases, you have a sound-producing (carrier) oscillator and a modulation oscillator (whether it is called LFO or not), and the sound-producing oscillator's frequency is increased or decreased according to the instantaneous value of the modulation oscillator's output.
However, there are some further restrictions needed in order to get good-sounding FM synthesis:
The output of the modulation oscillator must be symmetric — extending equally above and below zero. If this is not done, then increasing modulation depth (the amplitude of the frequency variation) will change the perceived pitch of the note as well as changing timbre, which is usually considered undesirable if you are trying to stick to a musical scale.
The change in frequency must be linear. This is sometimes referred to as "modulating frequency rather than pitch". This means that how far the carrier is modulated is equal up and down in Hz rather than in semitones. Again, if this is not done correctly then modulation will change the perceived pitch of the note.
Optional: The modulation oscillator's frequency should have a simple ratio with the carrier oscillator's unmodulated frequency. If the ratio is not simple, then the result is “inharmonic”: it contains frequency components that are not just integer multiples of the carrier frequency. These sounds will be heard to change in timbre as they play, quickly or slowly, and are often referred to as “bell-like”.
If you ignore some of these rules, you'll still get a frequency-modulated sound but it may not be as “conventionally musical”.
That's the big picture, now I'll comment on some of your wonderings:
The modulator as I understand in general synth applications adds an additional frequency to the carrier frequency.
This is not really true. In general, "modulation" means "change some parameter according to some other signal". In the case of FM, we change the main oscillator frequency, but there are lots of other parameters that could be modulated, such as amplitude, filter cutoff, wave shape, and so on.
Also, "add another frequency" is a rather ambiguous phrase. For example, if I take two oscillators at two different frequencies and add their output signals, then I have a signal to which I have "added another frequency", but that's not FM, that's just playing two notes at once. It's okay to say you "added" something when the signal flow is already established, but when we're trying to talk about what the signal flow is,, it's not very specific at all.
I wonder if I am able to create LFO effects with this modulator function evene if I have no explicit LFO functionality.
If you can turn the FM ratio low enough, then yes. (I'm assuming, being a digital FM synth, that it offers you a "FM ratio" control rather than a "modulator frequency" control.) But since in FM synthesis we usually want the modulator to stick to a certain ratio with the carrier, you'll find that when you play a higher note the rate of the modulation also increases (e.g. one octave up ⇒ twice as fast modulation), whereas a LFO would usually stick to the same low frequency.
Disclaimer: I'm fairly new to synthesis in a musical context myself, and haven't actually worked with digital FM synthesis, and some of my terminology may be a bit off. Suggestions welcome.